Questions — Edexcel (10514 questions)

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Edexcel M1 2001 June Q3
9 marks Moderate -0.8
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{218383c1-0875-46f2-9416-8e827065a7a6-3_540_1223_348_455}
\end{figure} A car of mass 1200 kg moves along a straight horizontal road. In order to obey a speed restriction, the brakes of the car are applied for 3 s , reducing the car's speed from \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The brakes are then released and the car continues at a constant speed of \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for a further 4 s . Figure 2 shows a sketch of the speed-time graph of the car during the 7 s interval. The graph consists of two straight line segments.
  1. Find the total distance moved by the car during this 7 s interval.
  2. Explain briefly how the speed-time graph shows that, when the brakes are applied, the car experiences a constant retarding force.
  3. Find the magnitude of this retarding force.
Edexcel M1 2001 June Q4
11 marks Standard +0.3
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{218383c1-0875-46f2-9416-8e827065a7a6-4_347_854_356_640}
\end{figure} A small parcel of mass 3 kg is held in equilibrium on a rough plane by the action of a horizontal force of magnitude 30 N acting in a vertical plane through a line of greatest slope. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Fig. 3. The parcel is modelled as a particle. The parcel is on the point of moving up the slope.
  1. Draw a diagram showing all the forces acting on the parcel.
  2. Find the normal reaction on the parcel.
  3. Find the coefficient of friction between the parcel and the plane.
Edexcel M1 2001 June Q5
13 marks Standard +0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{218383c1-0875-46f2-9416-8e827065a7a6-5_328_993_491_483}
\end{figure} A large \(\log A B\) is 6 m long. It rests in a horizontal position on two smooth supports \(C\) and \(D\), where \(A C = 1 \mathrm {~m}\) and \(B D = 1 \mathrm {~m}\), as shown in Figure 4. David needs an estimate of the weight of the log, but the log is too heavy to lift off both supports. When David applies a force of magnitude 1500 N vertically upwards to the \(\log\) at \(A\), the \(\log\) is about to tilt about \(D\).
  1. State the value of the reaction on the \(\log\) at \(C\) for this case. David initially models the log as uniform rod. Using this model,
  2. estimate the weight of the log The shape of the log convinces David that his initial modelling assumption is too simple. He removes the force at \(A\) and applies a force acting vertically upwards at \(B\). He finds that the log is about to tilt about \(C\) when this force has magnitude 1000 N. David now models the log as a non-uniform rod, with the distance of the centre of mass of the \(\log\) from \(C\) as \(x\) metres. Using this model, find
  3. a new estimate for the weight of the log,
  4. the value of \(x\).
  5. State how you have used the modeling assumption that the log is a rod.
Edexcel M1 2001 June Q6
13 marks Moderate -0.3
6. A breakdown van of mass 2000 kg is towing a car of mass 1200 kg along a straight horizontal road. The two vehicles are joined by a tow bar which remains parallel to the road. The van and the car experience constant resistances to motion of magnitudes 800 N and 240 N respectively. There is a constant driving force acting on the van of 2320 N . Find
  1. the magnitude of the acceleration of the van and the car,
  2. the tension in the tow bar. The two vehicles come to a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\). The driving force and the resistances to the motion are unchanged.
  3. Find the magnitude of the acceleration of the van and the car as they move up the hill and state whether their speed increases or decreases.
Edexcel M1 2001 June Q7
15 marks Standard +0.3
7. [In this question, the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and north respectively] A mountain rescue post \(O\) receives a distress call via a mobile phone from a walker who has broken a leg and cannot move. The walker says he is by a pipeline and he can also see a radio mast which he believes to be south-west of him. The pipeline is known to run north-south for a long distance through the point with position vector \(6 \mathbf { i } \mathrm {~km}\), relative to \(O\). The radio mast is known to be at the point with position vector \(2 \mathbf { j } \mathrm {~km}\), relative to \(O\).
  1. Using the information supplied by the walker, write down his position vector in the form \(( a \mathbf { i } + b \mathbf { j } )\). The rescue party moves at a horizontal speed of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). The leader of the party wants to give the walker and idea of how long it will take to for the rescue party to arrive.
  2. Calculate how long it will take for the rescue party to reach the walker's estimated position. The rescue party sets out and walks straight towards the walker's estimated position at a constant horizontal speed of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). After the party has travelled for one hour, the walker rings again. He is very apologetic and says that he now realises that the radio mask is in fact north-west of his position
  3. Find the position vector of the walker.
  4. Find in degrees to one decimal place, the bearing on which the rescue party should now travel in order to reach the walker directly. \section*{END}
Edexcel M1 2003 June Q1
6 marks Moderate -0.3
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d91990b5-b7ea-485c-aa4e-fe42b61ca7f8-2_302_807_379_603}
\end{figure} A uniform plank \(A B\) has mass 40 kg and length 4 m . It is supported in a horizontal position by two smooth pivots, one at the end \(A\), the other at the point \(C\) of the plank where \(A C = 3 \mathrm {~m}\), as shown in Fig. 1. A man of mass 80 kg stands on the plank which remains in equilibrium. The magnitudes of the reactions at the two pivots are each equal to \(R\) newtons. By modelling the plank as a rod and the man as a particle, find
  1. the value of \(R\),
  2. the distance of the man from \(A\).
    (4)
Edexcel M1 2003 June Q2
7 marks Easy -1.3
2. Two particles \(A\) and \(B\) have mass 0.12 kg and 0.08 kg respectively. They are initially at rest on a smooth horizontal table. Particle \(A\) is then given an impulse in the direction \(A B\) so that it moves with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) directly towards \(B\).
  1. Find the magnitude of this impulse, stating clearly the units in which your answer is given.
    (2) Immediately after the particles collide, the speed of \(A\) is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its direction of motion being unchanged.
  2. Find the speed of \(B\) immediately after the collision.
  3. Find the magnitude of the impulse exerted on \(A\) in the collision.
Edexcel M1 2003 June Q3
8 marks Moderate -0.8
3. A competitor makes a dive from a high springboard into a diving pool. She leaves the springboard vertically with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) upwards. When she leaves the springboard, she is 5 m above the surface of the pool. The diver is modelled as a particle moving vertically under gravity alone and it is assumed that she does not hit the springboard as she descends. Find
  1. her speed when she reaches the surface of the pool,
  2. the time taken to reach the surface of the pool.
  3. State two physical factors which have been ignored in the model.
Edexcel M1 2003 June Q4
8 marks Standard +0.3
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d91990b5-b7ea-485c-aa4e-fe42b61ca7f8-3_355_759_1087_605}
\end{figure} A parcel of mass 5 kg lies on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The parcel is held in equilibrium by the action of a horizontal force of magnitude 20 N , as shown in Fig. 2. The force acts in a vertical plane through a line of greatest slope of the plane. The parcel is on the point of sliding down the plane. Find the coefficient of friction between the parcel and the plane.
(8)
Edexcel M1 2003 June Q5
10 marks Moderate -0.3
5. A particle \(P\) moves with constant acceleration \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). At time \(t\) seconds, its velocity is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , \mathbf { v } = - 2 \mathbf { i } + 7 \mathbf { j }\).
  1. Find the value of \(t\) when \(P\) is moving parallel to the vector \(\mathbf { i }\).
  2. Find the speed of \(P\) when \(t = 3\).
  3. Find the angle between the vector \(\mathbf { j }\) and the direction of motion of \(P\) when \(t = 3\).
Edexcel M1 2003 June Q6
11 marks Moderate -0.3
6. A particle \(P\) of mass 3 kg is projected up a line of greatest slope of a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4 . The initial speed of P is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the frictional force acting on \(P\) as it moves up the plane,
  2. the distance moved by \(P\) up the plane before \(P\) comes to instantaneous rest.
Edexcel M1 2003 June Q7
12 marks Standard +0.3
7. Two trains \(A\) and \(B\) run on parallel straight tracks. Initially both are at rest in a station and level with each other. At time \(t = 0 , A\) starts to move. It moves with constant acceleration for 12 s up to a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and then moves at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Train \(B\) starts to move in the same direction as \(A\) when \(t = 40\), where \(t\) is measured in seconds. It accelerates with the same initial acceleration as \(A\), up to a speed of \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then moves at a constant speed of \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Train \(B\) overtakes \(A\) after both trains have reached their maximum speed. Train \(B\) overtakes \(A\) when \(t = T\).
  1. Sketch, on the same diagram, the speed-time graphs of both trains for \(0 \leq t \leq T\).
  2. Find the value of \(T\).
Edexcel M1 2003 June Q8
13 marks Moderate -0.3
8. A car which has run out of petrol is being towed by a breakdown truck along a straight horizontal road. The truck has mass 1200 kg and the car has mass 800 kg . The truck is connected to the car by a horizontal rope which is modelled as light and inextensible. The truck's engine provides a constant driving force of 2400 N . The resistances to motion of the truck and the car are modelled as constant and of magnitude 600 N and 400 N respectively. Find
  1. the acceleration of the truck and the ear,
  2. the tension in the rope. When the truck and car are moving at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the rope breaks. The engine of the truck provides the same driving force as before. The magnitude of the resistance to the motion of the truck remains 600 N .
  3. Show that the truck reaches a speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) approximately 6 s earlier than it would have done if the rope had not broken. \section*{END}
Edexcel M1 2006 June Q1
6 marks Easy -1.3
1. Figure 1 \includegraphics[max width=\textwidth, alt={}, center]{3a8395fd-6e44-48a1-8c97-3365a284956a-02_404_755_312_577} Figure 1 shows the speed-time graph of a cyclist moving on a straight road over a 7 s period. The sections of the graph from \(t = 0\) to \(t = 3\), and from \(t = 3\) to \(t = 7\), are straight lines. The section from \(t = 3\) to \(t = 7\) is parallel to the \(t\)-axis. State what can be deduced about the motion of the cyclist from the fact that
  1. the graph from \(t = 0\) to \(t = 3\) is a straight line,
  2. the graph from \(t = 3\) to \(t = 7\) is parallel to the \(t\)-axis.
  3. Find the distance travelled by the cyclist during this 7 s period.
Edexcel M1 2006 June Q2
7 marks Moderate -0.8
2. Two particles \(A\) and \(B\) have mass 0.4 kg and 0.3 kg respectively. They are moving in opposite directions on a smooth horizontal table and collide directly. Immediately before the collision, the speed of \(A\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As a result of the collision, the direction of motion of \(B\) is reversed and its speed immediately after the collision is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the speed of \(A\) immediately after the collision, stating clearly whether the direction of motion of \(A\) is changed by the collision,
  2. the magnitude of the impulse exerted on \(B\) in the collision, stating clearly the units in which your answer is given.
Edexcel M1 2006 June Q3
10 marks Moderate -0.8
3. A train moves along a straight track with constant acceleration. Three telegraph poles are set at equal intervals beside the track at points \(A , B\) and \(C\), where \(A B = 50 \mathrm {~m}\) and \(B C = 50 \mathrm {~m}\). The front of the train passes \(A\) with speed \(22.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and 2 s later it passes \(B\). Find
  1. the acceleration of the train,
  2. the speed of the front of the train when it passes \(C\),
  3. the time that elapses from the instant the front of the train passes \(B\) to the instant it passes \(C\).
Edexcel M1 2006 June Q4
11 marks Standard +0.3
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{3a8395fd-6e44-48a1-8c97-3365a284956a-05_273_611_319_676}
\end{figure} A particle \(P\) of mass 0.5 kg is on a rough plane inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). The particle is held at rest on the plane by the action of a force of magnitude 4 N acting up the plane in a direction parallel to a line of greatest slope of the plane, as shown in Figure 2. The particle is on the point of slipping up the plane.
  1. Find the coefficient of friction between \(P\) and the plane. The force of magnitude 4 N is removed.
  2. Find the acceleration of \(P\) down the plane.
Edexcel M1 2006 June Q5
13 marks Moderate -0.3
5.
[diagram]
A steel girder \(A B\) has weight 210 N . It is held in equilibrium in a horizontal position by two vertical cables. One cable is attached to the end \(A\). The other cable is attached to the point \(C\) on the girder, where \(A C = 90 \mathrm {~cm}\), as shown in Figure 3. The girder is modelled as a uniform rod, and the cables as light inextensible strings. Given that the tension in the cable at \(C\) is twice the tension in the cable at \(A\), find
  1. the tension in the cable at \(A\),
  2. show that \(A B = 120 \mathrm {~cm}\). A small load of weight \(W\) newtons is attached to the girder at \(B\). The load is modelled as a particle. The girder remains in equilibrium in a horizontal position. The tension in the cable at \(C\) is now three times the tension in the cable at \(A\).
  3. Find the value of \(W\).
Edexcel M1 2006 June Q6
13 marks Moderate -0.3
A car is towing a trailer along a straight horizontal road by means of a horizontal tow-rope. The mass of the car is 1400 kg . The mass of the trailer is 700 kg . The car and the trailer are modelled as particles and the tow-rope as a light inextensible string. The resistances to motion of the car and the trailer are assumed to be constant and of magnitude 630 N and 280 N respectively. The driving force on the car, due to its engine, is 2380 N . Find
  1. the acceleration of the car,
  2. the tension in the tow-rope. When the car and trailer are moving at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the tow-rope breaks. Assuming that the driving force on the car and the resistances to motion are unchanged,
  3. find the distance moved by the car in the first 4 s after the tow-rope breaks.
    (6)
  4. State how you have used the modelling assumption that the tow-rope is inextensible.
Edexcel M1 2006 June Q7
15 marks Moderate -0.3
  1. \hspace{0pt} [In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and north respectively.]
A ship \(S\) is moving with constant velocity \(( - 2.5 \mathbf { i } + 6 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). At time 1200, the position vector of \(S\) relative to a fixed origin \(O\) is \(( 16 \mathbf { i } + 5 \mathbf { j } )\) km. Find
  1. the speed of \(S\),
  2. the bearing on which \(S\) is moving. The ship is heading directly towards a submerged rock \(R\). A radar tracking station calculates that, if \(S\) continues on the same course with the same speed, it will hit \(R\) at the time 1500.
  3. Find the position vector of \(R\). The tracking station warns the ship's captain of the situation. The captain maintains \(S\) on its course with the same speed until the time is 1400 . He then changes course so that \(S\) moves due north at a constant speed of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Assuming that \(S\) continues to move with this new constant velocity, find
  4. an expression for the position vector of the ship \(t\) hours after 1400,
  5. the time when \(S\) will be due east of \(R\),
  6. the distance of \(S\) from \(R\) at the time 1600.
Edexcel M1 2007 June Q1
7 marks Moderate -0.8
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{5b5d70b1-1eb6-461f-9277-5912b914f443-02_579_490_301_730}
\end{figure} A particle \(P\) is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). A horizontal force of magnitude 12 N is applied to \(P\). The particle \(P\) is in equilibrium with the string taut and \(O P\) making an angle of \(20 ^ { \circ }\) with the downward vertical, as shown in Figure 1. Find
  1. the tension in the string,
  2. the weight of \(P\).
Edexcel M1 2007 June Q2
7 marks Moderate -0.3
2. Two particles \(A\) and \(B\), of mass 0.3 kg and \(m \mathrm {~kg}\) respectively, are moving in opposite directions along the same straight horizontal line so that the particles collide directly. Immediately before the collision, the speeds of \(A\) and \(B\) are \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. In the collision the direction of motion of each particle is reversed and, immediately after the collision, the speed of each particle is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the magnitude of the impulse exerted by \(B\) on \(A\) in the collision,
  2. the value of \(m\).
Edexcel M1 2007 June Q3
9 marks Moderate -0.3
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{5b5d70b1-1eb6-461f-9277-5912b914f443-04_282_842_296_561}
\end{figure} A uniform rod \(A B\) has length 1.5 m and mass 8 kg . A particle of mass \(m \mathrm {~kg}\) is attached to the rod at \(B\). The rod is supported at the point \(C\), where \(A C = 0.9 \mathrm {~m}\), and the system is in equilibrium with \(A B\) horizontal, as shown in Figure 2.
  1. Show that \(m = 2\). A particle of mass 5 kg is now attached to the rod at \(A\) and the support is moved from \(C\) to a point \(D\) of the rod. The system, including both particles, is again in equilibrium with \(A B\) horizontal.
  2. Find the distance \(A D\).
Edexcel M1 2007 June Q4
11 marks Moderate -0.8
A car is moving along a straight horizontal road. At time \(t = 0\), the car passes a point \(A\) with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car moves with constant speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) until \(t = 10 \mathrm {~s}\). The car then decelerates uniformly for 8 s . At time \(t = 18 \mathrm {~s}\), the speed of the car is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and this speed is maintained until the car reaches the point \(B\) at time \(t = 30 \mathrm {~s}\).
  1. Sketch, in the space below, a speed-time graph to show the motion of the car from \(A\) to \(B\). Given that \(A B = 526 \mathrm {~m}\), find
  2. the value of \(V\),
  3. the deceleration of the car between \(t = 10 \mathrm {~s}\) and \(t = 18 \mathrm {~s}\).
Edexcel M1 2007 June Q5
10 marks Moderate -0.8
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{5b5d70b1-1eb6-461f-9277-5912b914f443-08_218_479_287_744}
\end{figure} A small ring of mass 0.25 kg is threaded on a fixed rough horizontal rod. The ring is pulled upwards by a light string which makes an angle \(40 ^ { \circ }\) with the horizontal, as shown in Figure 3. The string and the rod are in the same vertical plane. The tension in the string is 1.2 N and the coefficient of friction between the ring and the rod is \(\mu\). Given that the ring is in limiting equilibrium, find
  1. the normal reaction between the ring and the rod,
  2. the value of \(\mu\).