7. A uniform sphere \(A\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal table when it collides directly with another uniform sphere \(B\) of mass \(2 m\) which is at rest on the table. The spheres are of equal radius and the coefficient of restitution between them is \(e\). The direction of motion of \(A\) is unchanged by the collision.
- Find the speeds of \(A\) and \(B\) immediately after the collision.
- Find the range of possible values of \(e\).
After being struck by \(A\), the sphere \(B\) collides directly with another sphere \(C\), of mass \(4 m\) and of the same size as \(B\). The sphere \(C\) is at rest on the table immediately before being struck by \(B\). The coefficient of restitution between \(B\) and \(C\) is also \(e\).
- Show that, after \(B\) has struck \(C\), there will be a further collision between \(A\) and \(B\).
\section*{Advanced/Advanced Subsidiary}
\section*{Wednesday 21 January 2004 - Afternoon}
Answer Book (AB16)
Mathematical Formulae (Lilac)
Graph Paper (ASG2)
Nil
Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G.
In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Mechanics M2), the paper reference (6678), your surname, other name and signature.
Whenever a numerical value of \(g\) is required, take \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
When a calculator is used, the answer should be given to an appropriate degree of accuracy.
A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has seven questions.
You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
- A car of mass 400 kg is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\). The resistance to motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons. When the car is moving at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the power developed by the car's engine is 10 kW .
Find the value of \(R\).
2. A particle \(P\) of mass 0.75 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of \(P\) is given by
$$\mathbf { v } = \left( t ^ { 2 } + 2 \right) \mathbf { i } - 6 t \mathbf { j } .$$ - Find the magnitude of \(\mathbf { F }\) when \(t = 4\).
When \(t = 5\), the particle \(P\) receives an impulse of magnitude \(9 \sqrt { } 2\) Ns in the direction of the vector \(\mathbf { i } - \mathbf { j }\).
- Find the velocity of \(P\) immediately after the impulse.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{69c60052-a23a-415a-b30f-3f5b85be2686-16_312_720_422_372}
\end{figure}
A particle \(P\) of mass 2 kg is projected from a point \(A\) up a line of greatest slope \(A B\) of a fixed plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal and \(A B = 3 \mathrm {~m}\) with \(B\) above \(A\), as shown in Fig. 1. The speed of \(P\) at \(A\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Assuming the plane is smooth, - find the speed of \(P\) at \(B\).
The plane is now assumed to be rough. At \(A\) the speed of \(P\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at \(B\) the speed of \(P\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By using the work-energy principle, or otherwise,
- find the coefficient of friction between \(P\) and the plane.
4.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{69c60052-a23a-415a-b30f-3f5b85be2686-16_679_643_347_1781}
\end{figure}
A uniform ladder, of weight \(W\) and length \(2 a\), rests in equilibrium with one end \(A\) on a smooth horizontal floor and the other end \(B\) on a rough vertical wall. The ladder is in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the ladder is \(\mu\). The ladder makes an angle \(\theta\) with the floor, where \(\tan \theta = 2\). A horizontal light inextensible string \(C D\) is attached to the ladder at the point \(C\), where \(A C = \frac { 1 } { 2 } a\). The string is attached to the wall at the point \(D\), with \(B D\) vertical, as shown in Fig. 2. The tension in the string is \(\frac { 1 } { 4 } W\). By modelling the ladder as a rod, - find the magnitude of the force of the floor on the ladder,
- show that \(\mu \geq \frac { 1 } { 2 }\).
- State how you have used the modelling assumption that the ladder is a rod.
5. A particle \(P\) is projected with velocity \(( 2 u \mathbf { i } + 3 u \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) from a point \(O\) on a horizontal plane, where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical unit vectors respectively. The particle \(P\) strikes the plane at the point \(A\) which is 735 m from \(O\). - Show that \(u = 24.5\).
- Find the time of flight from \(O\) to \(A\).
The particle \(P\) passes through a point \(B\) with speed \(65 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
- Find the height of \(B\) above the horizontal plane.
6. A smooth sphere \(A\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal table when it collides directly with another smooth sphere \(B\) of mass \(3 m\), which is at rest on the table. The coefficient of restitution between \(A\) and \(B\) is \(e\). The spheres have the same radius and are modelled as particles. - Show that the speed of \(B\) immediately after the collision is \(\frac { 1 } { 4 } ( 1 + e ) u\).
- Find the speed of \(A\) immediately after the collision.
Immediately after the collision the total kinetic energy of the spheres is \(\frac { 1 } { 6 } m u ^ { 2 }\).
- Find the value of \(e\).
- Hence show that \(A\) is at rest after the collision.
7.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{69c60052-a23a-415a-b30f-3f5b85be2686-17_291_661_388_1868}
\end{figure}
A loaded plate \(L\) is modelled as a uniform rectangular lamina \(A B C D\) and three particles. The sides \(C D\) and \(A D\) of the lamina have lengths \(5 a\) and \(2 a\) respectively and the mass of the lamina is \(3 m\). The three particles have mass \(4 m , m\) and \(2 m\) and are attached at the points \(A , B\) and \(C\) respectively, as shown in Fig. 3. - Show that the distance of the centre of mass of \(L\) from \(A D\) is \(2.25 a\).
- Find the distance of the centre of mass of \(L\) from \(A B\).
The point \(O\) is the mid-point of \(A B\). The loaded plate \(L\) is freely suspended from \(O\) and hangs at rest under gravity.
- Find, to the nearest degree, the size of the angle that \(A B\) makes with the horizontal.
A horizontal force of magnitude \(P\) is applied at \(C\) in the direction \(C D\). The loaded plate \(L\) remains suspended from \(O\) and rests in equilibrium with \(A B\) horizontal and \(C\) vertically below \(B\).
- Show that \(P = \frac { 5 } { 4 } \mathrm { mg }\).
- Find the magnitude of the force on \(L\) at \(O\).
END
\section*{Edexcel GCE
Mechanics M2
Advanced/Advanced Subsidiary
Friday 25 June 2004 - Morning
Time: \(\mathbf { 1 }\) hour \(\mathbf { 3 0 }\) minutes }
Answer Book (AB16)
Mathematical Formulae (Lilac)
Graph Paper (ASG2)
Candidates may use any calculator EXCEPT those with the facility for symbolic
Items included with question papers
Nil algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G.
In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Mechanics M2), the paper reference (6678), your surname, other name and signature.
Whenever a numerical value of \(g\) is required, take \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
When a calculator is used, the answer should be given to an appropriate degree of accuracy.
A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has seven questions.
You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
- A lorry of mass 1500 kg moves along a straight horizontal road. The resistance to the motion of the lorry has magnitude 750 N and the lorry's engine is working at a rate of 36 kW .
- Find the acceleration of the lorry when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
The lorry comes to a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 10 }\). The magnitude of the resistance to motion from non-gravitational forces remains 750 N .
The lorry moves up the hill at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). - Find the rate at which the lorry's engine is now working.
2. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]
A ball has mass 0.2 kg . It is moving with velocity ( \(30 \mathrm { i } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it is struck by a bat. The bat exerts an impulse of \(( - 4 \mathbf { i } + 4 \mathbf { j } )\) Ns on the ball.
Find - the velocity of the ball immediately after the impact,
- the angle through which the ball is deflected as a result of the impact,
- the kinetic energy lost by the ball in the impact.
3.
\section*{Figure 1}
\includegraphics[max width=\textwidth, alt={}]{69c60052-a23a-415a-b30f-3f5b85be2686-19_554_412_333_532}
Figure 1 shows a decoration which is made by cutting the shape of a simple tree from a sheet of uniform card. The decoration consists of a triangle \(A B C\) and a rectangle \(P Q R S\). The points \(P\) and \(S\) lie on \(B C\) and \(M\) is the mid-point of both \(B C\) and \(P S\). The triangle \(A B C\) is isosceles with \(A B = A C , B C = 4 \mathrm {~cm} , A M = 6 \mathrm {~cm} , P S = 2 \mathrm {~cm}\) and \(P Q = 3 \mathrm {~cm}\). - Find the distance of the centre of mass of the decoration from \(B C\).
The decoration is suspended from \(Q\) and hangs freely.
- Find, in degrees to one decimal place, the angle between \(P Q\) and the vertical.
4. At time \(t\) seconds, the velocity of a particle \(P\) is \([ ( 4 t - 7 ) \mathbf { i } - 5 \mathbf { j } ] \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , P\) is at the point with position vector \(( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\) relative to a fixed origin \(O\). - Find an expression for the position vector of \(P\) after \(t\) seconds, giving your answer in the form \(( a \mathbf { i } + b \mathbf { j } ) \mathrm { m }\).
A second particle \(Q\) moves with constant velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the position vector of \(Q\) is \(( - 7 \mathbf { i } ) \mathrm { m }\).
- Prove that \(P\) and \(Q\) collide.
5. Two small smooth spheres, \(P\) and \(Q\), of equal radius, have masses \(2 m\) and \(3 m\) respectively. The sphere \(P\) is moving with speed \(5 u\) on a smooth horizontal table when it collides directly with \(Q\), which is at rest on the table. The coefficient of restitution between \(P\) and \(Q\) is \(e\). - Show that the speed of \(Q\) immediately after the collision is \(2 ( 1 + e ) u\).
After the collision, \(Q\) hits a smooth vertical wall which is at the edge of the table and perpendicular to the direction of motion of \(Q\). The coefficient of restitution between \(Q\) and the wall is \(f , 0 < f \leq 1\).
- Show that, when \(e = 0.4\), there is a second collision between \(P\) and \(Q\).
Given that \(e = 0.8\) and there is a second collision between \(P\) and \(Q\),
- find the range of possible values of \(f\).
6. A uniform ladder \(A B\), of mass \(m\) and length \(2 a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is 0.6 . The other end \(B\) of the ladder rests against a smooth vertical wall.
A builder of mass \(10 m\) stands at the top of the ladder. To prevent the ladder from slipping, the builder's friend pushes the bottom of the ladder horizontally towards the wall with a force of magnitude \(P\). This force acts in a direction perpendicular to the wall. The ladder rests in equilibrium in a vertical plane perpendicular to the wall and makes an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac { 3 } { 2 }\). - Show that the reaction of the wall on the ladder has magnitude 7 mg .
- Find, in terms of \(m\) and \(g\), the range of values of \(P\) for which the ladder remains in equilibrium.
7.
\includegraphics[max width=\textwidth, alt={}, center]{69c60052-a23a-415a-b30f-3f5b85be2686-20_291_955_269_322}
In a ski-jump competition, a skier of mass 80 kg moves from rest at a point \(A\) on a ski-slope. The skier's path is an \(\operatorname { arc } A B\). The starting point \(A\) of the slope is 32.5 m above horizontal ground. The end \(B\) of the slope is 8.1 m above the ground. When the skier reaches \(B\), she is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and moving upwards at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Fig. 2. The distance along the slope from \(A\) to \(B\) is 60 m . The resistance to motion while she is on the slope is modelled as a force of constant magnitude \(R\) newtons. By using the work-energy principle, - find the value of \(R\).
On reaching \(B\), the skier then moves through the air and reaches the ground at the point \(C\). The motion of the skier in moving from \(B\) to \(C\) is modelled as that of a particle moving freely under gravity.
- Find the time for the skier to move from \(B\) to \(C\).
- Find the horizontal distance from \(B\) to \(C\).
- Find the speed of the skier immediately before she reaches \(C\).
\section*{Wednesday 12 January 2005 - Afternoon}
Items included with question papers
Nil
Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G.
In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Mechanics M2), the paper reference (6678), your surname, other name and signature.
Whenever a numerical value of \(g\) is required, take \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
When a calculator is used, the answer should be given to an appropriate degree of accuracy.
A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has seven questions.
The total mark for this paper is 75.
You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
1.
Figure 1
\includegraphics[max width=\textwidth, alt={}, center]{69c60052-a23a-415a-b30f-3f5b85be2686-21_460_632_251_406}
A uniform rod \(A B\), of length \(8 a\) and weight \(W\), is free to rotate in a vertical plane about a smooth pivot at \(A\). One end of a light inextensible string is attached to \(B\). The other end is attached to point \(C\) which is vertically above \(A\), with \(A C = 6 a\). The rod is in equilibrium with \(A B\) horizontal, as shown in Figure 1. - By taking moments about \(A\), or otherwise, show that the tension in the string is \(\frac { 5 } { 6 } W\).
- Calculate the magnitude of the horizontal component of the force exerted by the pivot on the rod.
2.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{69c60052-a23a-415a-b30f-3f5b85be2686-21_435_849_274_1772}
\end{figure}
Figure 2 shows a metal plate that is made by removing a circle of centre \(O\) and radius 3 cm from a uniform rectangular lamina \(A B C D\), where \(A B = 20 \mathrm {~cm}\) and \(B C = 10 \mathrm {~cm}\). The point \(O\) is 5 cm from both \(A B\) and \(C D\) and is 6 cm from \(A D\). - Calculate, to 3 significant figures, the distance of the centre of mass of the plate from \(A D\).
The plate is freely suspended from \(A\) and hangs in equilibrium.
- Calculate, to the nearest degree, the angle between \(A B\) and the vertical.
(3)
3.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{69c60052-a23a-415a-b30f-3f5b85be2686-22_330_587_281_473}
\end{figure}
A small package \(P\) is modelled as a particle of mass 0.6 kg . The package slides down a rough plane from a point \(S\) to a point \(T\), where \(S T = 12 \mathrm {~m}\). The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal and \(S T\) is a line of greatest slope of the plane, as shown in Figure 3. The speed of \(P\) at \(S\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(P\) at \(T\) is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate - the total loss of energy of \(P\) in moving from \(S\) to \(T\),
- the coefficient of friction between \(P\) and the plane.
4. A particle \(P\) of mass 0.4 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity of \(P , \mathrm { v } _ { \mathrm { m } } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by
$$\mathbf { v } = ( 6 t + 4 ) \mathbf { i } + \left( t ^ { 2 } + 3 t \right) \mathbf { j }$$
When \(t = 0 , P\) is at the point with position vector \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m }\). When \(t = 4 , P\) is at the point \(S\). - Calculate the magnitude of \(\mathbf { F }\) when \(t = 4\).
- Calculate the distance \(O S\).
5. A car of mass 1000 kg is towing a trailer of mass 1500 kg along a straight horizontal road. The tow-bar joining the car to the trailer is modelled as a light rod parallel to the road. The total resistance to motion of the car is modelled as having constant magnitude 750 N . The total resistance to motion of the trailer is modelled as of magnitude \(R\) newtons, where \(R\) is a constant. When the engine of the car is working at a rate of 50 kW , the car and the trailer travel at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). - Show that \(R = 1250\).
When travelling at \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the driver of the car disengages the engine and applies the brakes. The brakes provide a constant braking force of magnitude 1500 N to the car. The resisting forces of magnitude 750 N and 1250 N are assumed to remain unchanged. Calculate
- the deceleration of the car while braking,
- the thrust in the tow-bar while braking,
- the work done, in kJ , by the braking force in bringing the car and the trailer to rest.
- Suggest how the modelling assumption that the resistances to motion are constant could be refined to be more realistic.
6. A particle \(P\) of mass \(3 m\) is moving with speed \(2 u\) in a straight line on a smooth horizontal table. The particle \(P\) collides with a particle \(Q\) of mass \(2 m\) moving with speed \(u\) in the opposite direction to \(P\). The coefficient of restitution between \(P\) and \(Q\) is \(e\). - Show that the speed of \(Q\) after the collision is \(\frac { 1 } { 5 } u ( 9 e + 4 )\).
As a result of the collision, the direction of motion of \(P\) is reversed.
- Find the range of possible values of \(e\).
Given that the magnitude of the impulse of \(P\) on \(Q\) is \(\frac { 32 } { 5 } m u\),
- find the value of \(e\).
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 4}
\includegraphics[alt={},max width=\textwidth]{69c60052-a23a-415a-b30f-3f5b85be2686-23_392_1073_292_214}
\end{figure}
A particle \(P\) is projected from a point \(A\) with speed \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\alpha\), where \(\sin \alpha = \frac { 3 } { 5 }\). The point \(O\) is on horizontal ground, with \(O\) vertically below \(A\) and \(O A = 20 \mathrm {~m}\). The particle \(P\) moves freely under gravity and passes through a point \(B\), which is 16 m above ground, before reaching the ground at the point \(C\), as shown in Figure 4.
Calculate - the time of the flight from \(A\) to \(C\),
- the distance \(O C\),
- the speed of \(P\) at \(B\),
- the angle that the velocity of \(P\) at \(B\) makes with the horizontal.
Materials required for examination
Mathematical Formulae (Lilac or Green)
Items included with question papers Nil
Advanced/Advanced Subsidiary
Friday 24 June 2005 - Morning
Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G.
In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Mechanics M2), the paper reference (6678), your surname, other name and signature.
Whenever a numerical value of \(g\) is required, take \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
When a calculator is used, the answer should be given to an appropriate degree of accuracy.
A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has 7 questions.
The total mark for this paper is 75.
You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
- A car of mass 1200 kg moves along a straight horizontal road. The resistance to motion of the car from non-gravitational forces is of constant magnitude 600 N . The car moves with constant speed and the engine of the car is working at a rate of 21 kW .
- Find the speed of the car.
The car moves up a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 14 }\).
The car's engine continues to work at 21 kW and the resistance to motion from nongravitational forces remains of magnitude 600 N . - Find the constant speed at which the car can move up the hill.
2.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{69c60052-a23a-415a-b30f-3f5b85be2686-24_337_577_767_488}
\end{figure}
A thin uniform wire, of total length 20 cm , is bent to form a frame. The frame is in the shape of a trapezium \(A B C D\), where \(A B = A D = 4 \mathrm {~cm} , C D = 5 \mathrm {~cm}\) and \(A B\) is perpendicular to \(B C\) and \(A D\), as shown in Figure 1. - Find the distance of the centre of mass of the frame from \(A B\).
The frame has mass \(M\). A particle of mass \(k M\) is attached to the frame at \(C\). When the frame is freely suspended from the mid-point of \(B C\), the frame hangs in equilibrium with \(B C\) horizontal.
- Find the value of \(k\).
3. A particle \(P\) moves in a horizontal plane. At time \(t\) seconds, the position vector of \(P\) is \(\mathbf { r }\) metres relative to a fixed origin \(O\), and \(\mathbf { r }\) is given by
$$\mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } ,$$
where \(c\) is a positive constant. When \(t = 1.5\), the speed of \(P\) is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find - the value of \(c\),
- the acceleration of \(P\) when \(t = 1.5\).
4. A darts player throws darts at a dart board which hangs vertically. The motion of a dart is modelled as that of a particle moving freely under gravity. The darts move in a vertical plane which is perpendicular to the plane of the dart board. A dart is thrown horizontally with speed \(12.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It hits the board at a point which is 10 cm below the level from which it was thrown. - Find the horizontal distance from the point where the dart was thrown to the dart board.
The darts player moves his position. He now throws a dart from a point which is at a horizontal distance of 2.5 m from the board. He throws the dart at an angle of elevation \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 7 } { 24 }\). This dart hits the board at a point which is at the same level as the point from which it was thrown.
- Find the speed with which the dart is thrown.
5. Two small spheres \(A\) and \(B\) have mass \(3 m\) and \(2 m\) respectively. They are moving towards each other in opposite directions on a smooth horizontal plane, both with speed \(2 u\), when they collide directly. As a result of the collision, the direction of motion of \(B\) is reversed and its speed is unchanged. - Find the coefficient of restitution between the spheres.
Subsequently, \(B\) collides directly with another small sphere \(C\) of mass \(5 m\) which is at rest. The coefficient of restitution between \(B\) and \(C\) is \(\frac { 3 } { 5 }\).
- Show that, after \(B\) collides with \(C\), there will be no further collisions between the spheres.
(7)
6.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{69c60052-a23a-415a-b30f-3f5b85be2686-25_392_622_255_445}
\end{figure}
A uniform pole \(A B\), of mass 30 kg and length 3 m , is smoothly hinged to a vertical wall at one end \(A\). The pole is held in equilibrium in a horizontal position by a light rod \(C D\). One end \(C\) of the rod is fixed to the wall vertically below \(A\). The other end \(D\) is freely jointed to the pole so that \(\angle A C D = 30 ^ { \circ }\) and \(A D = 0.5 \mathrm {~m}\), as shown in Figure 2. Find - the thrust in the rod CD,
- the magnitude of the force exerted by the wall on the pole at \(A\).
The rod \(C D\) is removed and replaced by a longer light rod \(C M\), where \(M\) is the mid-point of \(A B\). The rod is freely jointed to the pole at \(M\). The pole \(A B\) remains in equilibrium in a horizontal position.
- Show that the force exerted by the wall on the pole at \(A\) now acts horizontally.
(2)
7. At a demolition site, bricks slide down a straight chute into a container. The chute is rough and is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The distance travelled down the chute by each brick is 8 m . A brick of mass 3 kg is released from rest at the top of the chute. When it reaches the bottom of the chute, its speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). - Find the potential energy lost by the brick in moving down the chute.
- By using the work-energy principle, or otherwise, find the constant frictional force acting on the brick as it moves down the chute.
- Hence find the coefficient of friction between the brick and the chute.
Another brick of mass 3 kg slides down the chute. This brick is given an initial speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the chute.
- Find the speed of this brick when it reaches the bottom of the chute.
(5)
\section*{Advanced /Advanced Subsidiary}
Mathematical Formulae (Green or Lilac)
\section*{Thursday 12 January 2006 - Afternoon}
Nil
Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G.
In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Mechanics M2), the paper reference (6678), your surname, other name and signature.
Whenever a numerical value of \(g\) is required, take \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
When a calculator is used, the answer should be given to an appropriate degree of accuracy.
A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2). There are 7 questions on this paper. The total mark for this paper is 75 .
You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
- A brick of mass 3 kg slides in a straight line on a horizontal floor. The brick is modelled as a particle and the floor as a rough plane. The initial speed of the brick is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The brick is brought to rest after moving 12 m by the constant frictional force between the brick and the floor.
- Calculate the kinetic energy lost by the brick in coming to rest, stating the units of your answer.
- Calculate the coefficient of friction between the brick and the floor.
- A particle \(P\) of mass 0.4 kg is moving so that its position vector \(\mathbf { r }\) metres at time \(t\) seconds is given by
$$\mathbf { r } = \left( t ^ { 2 } + 4 t \right) \mathbf { i } + \left( 3 t - t ^ { 3 } \right) \mathbf { j } .$$ - Calculate the speed of \(P\) when \(t = 3\).
When \(t = 3\), the particle \(P\) is given an impulse \(( 8 \mathbf { i } - 12 \mathbf { j } ) \mathrm { N } \mathrm { s }\).
- Find the velocity of \(P\) immediately after the impulse.
3. A car of mass 1000 kg is moving along a straight horizontal road. The resistance to motion is modelled as a constant force of magnitude \(R\) newtons. The engine of the car is working at a rate of 12 kW . When the car is moving with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the car is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). - Show that \(R = 600\).
The car now moves with constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downhill on a straight road inclined at \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 40 }\). The engine of the car is now working at a rate of 7 kW . The resistance to motion from non-gravitational forces remains of magnitude \(R\) newtons.
- Calculate the value of \(U\).
4. A particle \(A\) of mass \(2 m\) is moving with speed \(3 u\) in a straight line on a smooth horizontal table. The particle collides directly with a particle \(B\) of mass \(m\) moving with speed \(2 u\) in the opposite direction to \(A\). Immediately after the collision the speed of \(B\) is \(\frac { 8 } { 3 } u\) and the direction of motion of \(B\) is reversed. - Calculate the coefficient of restitution between \(A\) and \(B\).
- Show that the kinetic energy lost in the collision is \(7 m u ^ { 2 }\).
After the collision \(B\) strikes a fixed vertical wall that is perpendicular to the direction of motion of \(B\). The magnitude of the impulse of the wall on \(B\) is \(\frac { 14 } { 3 } m u\).
- Calculate the coefficient of restitution between \(B\) and the wall.
5.
Figure 1
\includegraphics[max width=\textwidth, alt={}, center]{69c60052-a23a-415a-b30f-3f5b85be2686-27_540_846_226_1683}
Figure 1 shows a triangular lamina \(A B C\). The coordinates of \(A , B\) and \(C\) are ( 0,4 ), ( 9,0 ) and \(( 0 , - 4 )\) respectively. Particles of mass \(4 m , 6 m\) and \(2 m\) are attached at \(A , B\) and \(C\) respectively. - Calculate the coordinates of the centre of mass of the three particles, without the lamina.
(4)
The lamina \(A B C\) is uniform and of mass \(k m\). The centre of mass of the combined system consisting of the three particles and the lamina has coordinates \(( 4 , \lambda )\). - Show that \(k = 6\).
- Calculate the value of \(\lambda\).
The combined system is freely suspended from \(O\) and hangs at rest.
- Calculate, in degrees to one decimal place, the angle between \(A C\) and the vertical.
(3)
\section*{6.}
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{69c60052-a23a-415a-b30f-3f5b85be2686-28_478_380_274_532}
\end{figure}
A ladder \(A B\), of weight \(W\) and length \(4 a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is \(\mu\). The other end \(B\) rests against a smooth vertical wall. The ladder makes an angle \(\theta\) with the horizontal, where \(\tan \theta = 2\). A load of weight \(4 W\) is placed at the point \(C\) on the ladder, where \(A C = 3 a\), as shown in Figure 2. The ladder is modelled as a uniform rod which is in a vertical plane perpendicular to the wall. The load is modelled as a particle. Given that the system is in limiting equilibrium, - show that \(\mu = 0.35\).
A second load of weight \(k W\) is now placed on the ladder at \(A\). The load of weight \(4 W\) is removed from \(C\) and placed on the ladder at \(B\). The ladder is modelled as a uniform rod which is in a vertical plane perpendicular to the wall. The loads are modelled as particles. Given that the ladder and the loads are in equilibrium,
- find the range of possible values of \(k\).
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{69c60052-a23a-415a-b30f-3f5b85be2686-28_335_862_262_1729}
\end{figure}
The object of a game is to throw a ball \(B\) from a point \(A\) to hit a target \(T\) which is placed at the top of a vertical pole, as shown in Figure 3. The point \(A\) is 1 m above horizontal ground and the height of the pole is 2 m . The pole is at a horizontal distance of 10 m from \(A\). The ball \(B\) is projected from \(A\) with a speed of \(11 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(30 ^ { \circ }\). The ball hits the pole at the point \(C\). The ball \(B\) and the target \(T\) are modelled as particles. - Calculate, to 2 decimal places, the time taken for \(B\) to move from \(A\) to \(C\).
- Show that \(C\) is approximately 0.63 m below \(T\).
The ball is thrown again from \(A\). The speed of projection of \(B\) is increased to \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the angle of elevation remaining \(30 ^ { \circ }\). This time \(B\) hits \(T\).
- Calculate the value of \(V\).
- Explain why, in practice, a range of values of \(V\) would result in \(B\) hitting the target.
END
Mathematical Formulae (Lilac or Green)
\section*{Tuesday 6 June 2006 - Afternoon}
Nil
Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G.
In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Mechanics M2), the paper reference (6678), your surname, other name and signature.
Whenever a numerical value of \(g\) is required, take \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
When a calculator is used, the answer should be given to an appropriate degree of accuracy.
A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has 8 questions.
The total mark for this paper is 75 .
You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
- A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, its acceleration is \(( 5 - 2 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\), measured in the direction of \(x\) increasing. When \(t = 0\), its velocity is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) measured in the direction of \(x\) increasing. Find the time when \(P\) is instantaneously at rest in the subsequent motion.
- A car of mass 1200 kg moves along a straight horizontal road with a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion of the car has magnitude 600 N .
- Find, in kW , the rate at which the engine of the car is working.
The car now moves up a hill inclined at \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 28 }\). The resistance to motion of the car from non-gravitational forces remains of magnitude 600 N . The engine of the car now works at a rate of 30 kW . - Find the acceleration of the car when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. A cricket ball of mass 0.5 kg is struck by a bat. Immediately before being struck, the velocity of the ball is \(( - 30 \mathbf { i } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after being struck, the velocity of the ball is \(( 16 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). - Find the magnitude of the impulse exerted on the ball by the bat.
In the subsequent motion, the position vector of the ball is \(\mathbf { r }\) metres at time \(t\) seconds. In a model of the situation, it is assumed that \(\mathbf { r } = \left[ 16 t \mathbf { i } + \left( 20 t - 5 t ^ { 2 } \right) \mathbf { j } \right]\). Using this model,
- find the speed of the ball when \(t = 3\).
(4)
4.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{69c60052-a23a-415a-b30f-3f5b85be2686-30_398_643_274_461}
\end{figure}
Figure 1 shows four uniform rods joined to form a rigid rectangular framework \(A B C D\), where \(A B = C D = 2 a\), and \(B C = A D = 3 a\). Each rod has mass \(m\). Particles, of mass \(6 m\) and \(2 m\), are attached to the framework at points \(C\) and \(D\) respectively. - Find the distance of the centre of mass of the loaded framework from
- \(A B\),
- \(A D\).
The loaded framework is freely suspended from \(B\) and hangs in equilibrium.
- Find the angle which \(B C\) makes with the vertical.
5. A vertical cliff is 73.5 m high. Two stones \(A\) and \(B\) are projected simultaneously. Stone \(A\) is projected horizontally from the top of the cliff with speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Stone \(B\) is projected from the bottom of the cliff with speed \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal. The stones move freely under gravity in the same vertical plane and collide in mid-air. By considering the horizontal motion of each stone, - prove that \(\cos \alpha = \frac { 4 } { 5 }\).
- Find the time which elapses between the instant when the stones are projected and the instant when they collide.
6.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{69c60052-a23a-415a-b30f-3f5b85be2686-30_275_855_239_1736}
\end{figure}
A wooden plank \(A B\) has mass \(4 m\) and length \(4 a\). The end \(A\) of the plank lies on rough horizontal ground. A small stone of mass \(m\) is attached to the plank at \(B\). The plank is resting on a small smooth horizontal peg \(C\), where \(B C = a\), as shown in Figure 2. The plank is in equilibrium making an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between the plank and the ground is \(\mu\). The plank is modelled as a uniform rod lying in a vertical plane perpendicular to the peg, and the stone as a particle. Show that - the reaction of the peg on the plank has magnitude \(\frac { 16 } { 5 } \mathrm { mg }\),
- \(\mu \geq \frac { 48 } { 61 }\).
- State how you have used the information that the peg is smooth.
7. A particle \(P\) has mass 4 kg . It is projected from a point \(A\) up a line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between \(P\) and the plane is \(\frac { 2 } { 7 }\). The particle comes to rest instantaneously at the point \(B\) on the plane, where \(A B = 2.5 \mathrm {~m}\). It then moves back down the plane to \(A\). - Find the work done by friction as \(P\) moves from \(A\) to \(B\).
- Using the work-energy principle, find the speed with which \(P\) is projected from \(A\).
- Find the speed of \(P\) when it returns to \(A\).