3.02h Motion under gravity: vector form

1. The centre of the Earth is the point \(O\) and the Earth is modelled as a fixed sphere of radius \(R\).
   At time \(t = 0\), a particle \(P\) is projected vertically upwards with speed \(U\) from a point \(A\) on the surface of the Earth.

At time \(t\) seconds, where \(t \geqslant 0\)

• \(\quad P\) is a distance \(x\) from \(O\)
• \(P\) is moving with speed \(v\)
• \(P\) has acceleration of magnitude \(\frac { g R ^ { 2 } } { x ^ { 2 } }\) directed towards \(O\)

Air resistance is modelled as being negligible.

1. Show that \(v ^ { 2 } = \frac { 2 g R ^ { 2 } } { x } + U ^ { 2 } - 2 g R\) Particle \(P\) is first moving with speed \(\frac { 1 } { 2 } \sqrt { g R }\) at the point \(B\).
2. Given that \(U = \sqrt { g R }\) find, in terms of \(R\), the distance \(A B\).
3. Find, in terms of \(g\) and \(R\), the smallest value of \(U\) that would ensure that \(P\) never comes to rest, explaining your reasoning.

6. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held at rest with the string taut and horizontal and is then projected vertically downwards with speed \(u\), as shown in Figure 5. Air resistance is modelled as being negligible.
At the instant when the string has turned through an angle \(\theta\) and the string is taut, the tension in the string is \(T\).

1. Show that \(T = \frac { m u ^ { 2 } } { a } + 3 m g \sin \theta\) Given that \(u = 2 \sqrt { \frac { 3 a g } { 5 } }\)
2. find, in terms of \(a\) and \(g\), the speed of \(P\) at the instant when the string goes slack.
3. Hence find, in terms of \(a\), the maximum height of \(P\) above \(O\) in the subsequent motion.

6. \begin{figure}[h] \end{figure} A fixed solid sphere has centre \(O\) and radius \(r\).
A particle \(P\) of mass \(m\) is held at rest on the smooth surface of the sphere at \(A\), the highest point of the sphere.
The particle \(P\) is then projected horizontally from \(A\) with speed \(u\) and moves on the surface of the sphere.
At the instant when \(P\) reaches the point \(B\) on the sphere, where angle \(A O B = \theta , P\) is moving with speed \(v\), as shown in Figure 4. At this instant, \(P\) loses contact with the surface of the sphere.

1. Show that $$\cos \theta = \frac { 2 g r + u ^ { 2 } } { 3 g r }$$ In the subsequent motion, the particle \(P\) crosses the horizontal through \(O\) at the point \(C\), also shown in Figure 4. At the instant \(P\) passes through \(C , P\) is moving at an angle \(\alpha\) to the horizontal.
   Given that \(u ^ { 2 } = \frac { 2 g r } { 5 }\)
2. find the exact value of \(\tan \alpha\).

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a point \(O\). The point \(A\) is vertically below \(O\), and \(O A = a\). The particle is projected horizontally from \(A\) with speed \(\sqrt { } ( 3 a g )\). When \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\) and the string is still taut, the tension in the string is \(T\) and the speed of \(P\) is \(v\), as shown in Figure 2.

1. Find, in terms of \(a , g\) and \(\theta\), an expression for \(v ^ { 2 }\).
2. Show that \(T = ( 1 - 3 \cos \theta ) m g\). The string becomes slack when \(P\) is at the point \(B\).
3. Find, in terms of \(a\), the vertical height of \(B\) above \(A\). After the string becomes slack, the highest point reached by \(P\) is \(C\).
4. Find, in terms of \(a\), the vertical height of \(C\) above \(B\).

3. Above the earth's surface, the magnitude of the force on a particle due to the earth's gravity is inversely proportional to the square of the distance of the particle from the centre of the earth. Assuming that the earth is a sphere of radius \(R\), and taking \(g\) as the acceleration due to gravity at the surface of the earth,

1. prove that the magnitude of the gravitational force on a particle of mass \(m\) when it is a distance \(x ( x \geq R )\) from the centre of the earth is \(\frac { m g R ^ { 2 } } { x ^ { 2 } }\). A particle is fired vertically upwards from the surface of the earth with initial speed \(u\), where \(u ^ { 2 } = \frac { 3 } { 2 } g R\). Ignoring air resistance,
2. find, in terms of \(g\) and \(R\), the speed of the particle when it is at a height \(2 R\) above the surface of the earth.

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is fixed at a point \(O\). The particle is held with the string taut and \(O P\) horizontal. It is then projected vertically downwards with speed \(u\), where \(u ^ { 2 } = \frac { 3 } { 2 } g a\). When \(O P\) has turned through an angle \(\theta\) and the string is still taut, the speed of \(P\) is \(v\) and the tension in the string is \(T\), as shown in Fig. 3.

1. Find an expression for \(v ^ { 2 }\) in terms of \(a , g\) and \(\theta\).
2. Find an expression for \(T\) in terms of \(m , g\) and \(\theta\).
3. Prove that the string becomes slack when \(\theta = 210 ^ { \circ }\).
4. State, with a reason, whether \(P\) would complete a vertical circle if the string were replaced by a light rod. After the string becomes slack, \(P\) moves freely under gravity and is at the same level as \(O\) when it is at the point \(A\).
5. Explain briefly why the speed of \(P\) at \(A\) is \(\sqrt { } \left( \frac { 3 } { 2 } g a \right)\). The direction of motion of \(P\) at \(A\) makes an angle \(\varphi\) with the horizontal.
6. Find \(\varphi\).

6. One end of a light inextensible string of length \(l\) is attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\) which is hanging freely at rest at point \(B\). The particle \(P\) is projected horizontally from \(B\) with speed \(\sqrt { } ( 3 g l )\). When \(A P\) makes an angle \(\theta\) with the downward vertical and the string remains taut, the tension in the string is \(T\).

1. Show that \(T = m g ( 1 + 3 \cos \theta )\).
2. Find the speed of \(P\) at the instant when the string becomes slack.
3. Find the maximum height above the level of \(B\) reached by \(P\).

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a point \(O\). The point \(A\) is vertically below \(O\), and \(O A = a\). The particle is projected horizontally from \(A\) with speed \(\sqrt { } ( 3 a g )\). When \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\) and the string is still taut, the tension in the string is \(T\) and the speed of \(P\) is \(v\), as shown in Figure 2.

1. Find, in terms of \(a , g\) and \(\theta\), an expression for \(v ^ { 2 }\).
2. Show that \(T = ( 1 - 3 \cos \theta ) m g\). The string becomes slack when \(P\) is at the point \(B\).
3. Find, in terms of \(a\), the vertical height of \(B\) above \(A\). After the string becomes slack, the highest point reached by \(P\) is \(C\).
4. Find, in terms of \(a\), the vertical height of \(C\) above \(B\).

6. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). At time \(t = 0 , P\) is projected vertically downwards with speed \(\sqrt { } \left( \frac { 5 } { 2 } g a \right)\) from a point \(A\) which is at the same level as \(O\) and a distance \(a\) from \(O\). When the string has turned through an angle \(\theta\) and the string is still taut, the speed of \(P\) is \(v\) and the tension in the string is \(T\), as shown in Figure 2.

1. Show that \(v ^ { 2 } = \frac { g a } { 2 } ( 5 + 4 \sin \theta )\).
2. Find \(T\) in terms of \(m , g\) and \(\theta\). The string becomes slack when \(\theta = \alpha\).
3. Find the value of \(\alpha\). The particle is projected again from \(A\) with the same velocity as before. When \(P\) is at the same level as \(O\) for the first time after leaving \(A\), the string meets a small smooth peg \(B\) which has been fixed at a distance \(\frac { 1 } { 2 } a\) from \(O\). The particle now moves on an arc of a circle centre \(B\). Given that the particle reaches the point \(C\), which is \(\frac { 1 } { 2 } a\) vertically above the point \(B\), without the string going slack,
4. find the tension in the string when \(P\) is at the point \(C\).

7. \begin{figure}[h] \end{figure} A particle is projected from the highest point \(A\) on the outer surface of a fixed smooth sphere of radius \(a\) and centre \(O\). The lowest point \(B\) of the sphere is fixed to a horizontal plane. The particle is projected horizontally from \(A\) with speed \(\frac { 1 } { 2 } \sqrt { } ( g a )\). The particle leaves the surface of the sphere at the point \(C\), where \(\angle A O C = \theta\), and strikes the plane at the point \(P\), as shown in Figure 5.

1. Show that \(\cos \theta = \frac { 3 } { 4 }\).
2. Find the angle that the velocity of the particle makes with the horizontal as it reaches \(P\).

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(O\). The particle is hanging in equilibrium at the point \(A\), vertically below \(O\), when it is set in motion with a horizontal speed \(\frac { 1 } { 2 } \sqrt { } ( 11 g l )\). When the string has turned through an angle \(\theta\) and the string is still taut, the tension in the string is \(T\).

1. Show that \(T = 3 m g \left( \cos \theta + \frac { 1 } { 4 } \right)\). At the instant when \(P\) reaches the point \(B\), the string becomes slack. Find
2. the speed of \(P\) at \(B\),
3. the maximum height above \(B\) reached by \(P\) before it starts to fall.
   

5. A particle \(P\) is moving in a straight line with simple harmonic motion on a smooth horizontal floor. The particle comes to instantaneous rest at points \(A\) and \(B\) where \(A B\) is 0.5 m . The mid-point of \(A B\) is \(O\). The mid-point of \(O A\) is \(C\). The mid-point of \(O B\) is \(D\). The particle takes 0.2 s to travel directly from \(C\) to \(D\). At time \(t = 0 , P\) is moving through \(O\) towards \(A\).

1. Show that the period of the motion is \(\frac { 6 } { 5 } \mathrm {~s}\).
2. Find the distance of \(P\) from \(B\) when \(t = 2 \mathrm {~s}\).
3. Find the maximum magnitude of the acceleration of \(P\).
4. Find the maximum speed of \(P\).

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is fixed at the point \(O\). The particle is initially held with \(O P\) horizontal and the string taut. It is then projected vertically upwards with speed \(u\), where \(u ^ { 2 } = 5 a g\). When \(O P\) has turned through an angle \(\theta\) the speed of \(P\) is \(v\) and the tension in the string is \(T\), as shown in Figure 5.

1. Find, in terms of \(a , g\) and \(\theta\), an expression for \(v ^ { 2 }\).
2. Find, in terms of \(m , g\) and \(\theta\), an expression for \(T\).
3. Prove that \(P\) moves in a complete circle.
4. Find the maximum speed of \(P\).

A light elastic string, of natural length \(3 a\) and modulus of elasticity \(6 m g\), has one end attached to a fixed point \(A\). A particle \(P\) of mass \(2 m\) is attached to the other end of the string and hangs in equilibrium at the point \(O\), vertically below \(A\).

1. Find the distance \(A O\). The particle is now raised to point \(C\) vertically below \(A\), where \(A C > 3 a\), and is released from rest.
2. Show that \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { } \left( \frac { a } { g } \right)\). It is given that \(O C = \frac { 1 } { 4 } a\).
3. Find the greatest speed of \(P\) during the motion. The point \(D\) is vertically above \(O\) and \(O D = \frac { 1 } { 8 } a\). The string is cut as \(P\) passes through \(D\), moving upwards.
4. Find the greatest height of \(P\) above \(O\) in the subsequent motion.
   

6. \begin{figure}[h] \end{figure} A particle \(P\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is held at the point \(A\), where \(O A = a\) and \(O A\) is horizontal. The point \(B\) is vertically above \(O\) and the point \(C\) is vertically below \(O\), with \(O B = O C = a\), as shown in Figure 5. The particle is projected vertically upwards with speed \(3 \sqrt { } ( a g )\).

1. Show that \(P\) will pass through \(B\).
2. Find the speed of \(P\) as it reaches \(C\). As \(P\) passes through \(C\) it receives an impulse. Immediately after this, the speed of \(P\) is \(\frac { 5 } { 12 } \sqrt { } ( 11 a g )\) and the direction of motion of \(P\) is unchanged.
3. Find the angle between the string and the downward vertical when \(P\) comes to instantaneous rest.

A solid smooth sphere, with centre \(O\) and radius \(r\), is fixed to a point \(A\) on a horizontal floor. A particle \(P\) is placed on the surface of the sphere at the point \(B\), where \(B\) is vertically above \(A\). The particle is projected horizontally from \(B\) with speed \(\frac { \sqrt { g r } } { 2 }\) and starts to move on the surface of the sphere. When \(O P\) makes an angle \(\theta\) with the upward vertical and \(P\) remains in contact with the sphere, the speed of \(P\) is \(v\).

1. Show that \(v ^ { 2 } = \frac { g r } { 4 } ( 9 - 8 \cos \theta )\). The particle leaves the surface of the sphere when \(\theta = \alpha\).
2. Find the value of \(\cos \alpha\). After leaving the surface of the sphere, \(P\) moves freely under gravity and hits the floor at the point \(C\). Given that \(r = 0.5 \mathrm {~m}\),
3. find, to 2 significant figures, the distance \(A C\).

7. A smooth solid hemisphere is fixed with its plane face on a horizontal table and its curved surface uppermost. The plane face of the hemisphere has centre \(O\) and radius \(a\). The point \(A\) is the highest point on the hemisphere. A particle \(P\) is placed on the hemisphere at \(A\). It is then given an initial horizontal speed \(u\), where \(u ^ { 2 } = \frac { 1 } { 2 } ( a g )\). When \(O P\) makes an angle \(\theta\) with \(O A\), and while \(P\) remains on the hemisphere, the speed of \(P\) is \(v\).

1. Find an expression for \(v ^ { 2 }\).
2. Show that, when \(\theta = \arccos 0.9 , P\) is still on the hemisphere.
3. Find the value of \(\cos \theta\) when \(P\) leaves the hemisphere.
4. Find the value of \(v\) when \(P\) leaves the hemisphere. After leaving the hemisphere \(P\) strikes the table at \(B\).
5. Find the speed of \(P\) at \(B\).
6. Find the angle at which \(P\) strikes the table. \section*{Alternative Question 2:}
   1. Two light elastic strings \(A B\) and \(B C\) are joined at \(B\). The string \(A B\) has natural length 1 m and modulus of elasticity 15 N . The string \(B C\) has natural length 1.2 m and modulus of elasticity 30 N . The ends \(A\) and \(C\) are attached to fixed points 3 m apart and the strings rest in equilibrium with \(A B C\) in a straight line.
   Find the tension in the combined string \(A C\).

7. \begin{figure}[h] \end{figure} At time \(t = 0\), a particle \(P\) of mass 0.7 kg is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a fixed point \(O\) at an angle \(\theta ^ { \circ }\) to the horizontal. The particle moves freely under gravity. At time \(t = 2\) seconds, \(P\) passes through the point \(A\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving downwards at \(45 ^ { \circ }\) to the horizontal, as shown in Figure 4. Find

1. the value of \(\theta\),
2. the kinetic energy of \(P\) as it reaches the highest point of its path. For an interval of \(T\) seconds, the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is such that \(v \leqslant 6\)
3. Find the value of \(T\).

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.

1. 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.
2. Find the speed with which the dart is thrown.

1 A particle \(P\) is projected vertically downwards from a fixed point \(O\) with initial speed \(4.2 \mathrm {~ms} ^ { - 1 }\), and takes 1.5 s to reach the ground. Calculate

1. the speed of \(P\) when it reaches the ground,
2. the height of \(O\) above the ground,
3. the speed of \(P\) when it is 5 m above the ground.

3 A particle is projected vertically upwards with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 2.5 m above the ground.

1. Calculate the speed of the particle when it strikes the ground.
2. Calculate the time after projection when the particle reaches the ground.
3. Sketch on separate diagrams
   1. the \(( t , v )\) graph,
   2. the \(( t , x )\) graph,
      representing the motion of the particle.

5 \includegraphics[max width=\textwidth, alt={}, center]{4c6c9323-8238-4ec2-94a1-6e8188a34521-03_538_917_918_614} \(X\) is a point on a smooth plane inclined at \(\theta ^ { \circ }\) to the horizontal. \(Y\) is a point directly above the line of greatest slope passing through \(X\), and \(X Y\) is horizontal. A particle \(P\) is projected from \(X\) with initial speed \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down the line of greatest slope, and simultaneously a particle \(Q\) is released from rest at \(Y\). \(P\) moves with acceleration \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and \(Q\) descends freely under gravity (see diagram). The two particles collide at the point on the plane directly below \(Y\) at time \(T\) s after being set in motion.

1. (a) Express in terms of \(T\) the distances travelled by the particles before the collision.
   (b) Calculate \(\theta\).
   (c) Using the answers to parts (a) and (b), show that \(T = \frac { 2 } { 3 }\).
2. Calculate the speeds of the particles immediately before they collide.

2 A particle is projected vertically upwards with speed \(7 \mathrm {~ms} ^ { - 1 }\) from a point on the ground.

1. Find the speed of the particle and its distance above the ground 0.4 s after projection.
2. Find the total distance travelled by the particle in the first 0.9 s after projection.

2 A particle \(P\) is projected vertically upwards and reaches its greatest height 0.5 s after the instant of projection. Calculate

1. the speed of projection of \(P\),
2. the greatest height of \(P\) above the point of projection. It is given that the point of projection is 0.539 m above the ground.
3. Find the speed of \(P\) immediately before it strikes the ground.

5 A particle \(P\) is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) from the top of a smooth inclined plane of length \(2 d\) metres. After its projection \(P\) moves downwards along a line of greatest slope with acceleration \(4 \mathrm {~ms} ^ { - 2 }\). At the instant 3 s after projection \(P\) has moved half way down the plane. \(P\) reaches the foot of the plane 5 s after the instant of projection.

1. Form two simultaneous equations in \(u\) and \(d\), and hence calculate the speed of projection of \(P\) and the length of the plane.
2. Find the inclination of the plane to the horizontal.
3. Given that the contact force exerted on \(P\) by the plane has magnitude 6 N , calculate the mass of \(P\).