6.03f Impulse-momentum: relation

1. A particle \(P\) of mass 0.75 kg is moving with velocity \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse \(( 6 \mathbf { i } + 6 \mathbf { j } ) \mathrm { Ns }\). The angle between the velocity of \(P\) before the impulse and the velocity of \(P\) after the impulse is \(\theta ^ { \circ }\).

Find

1. the value of \(\theta\),
2. the kinetic energy gained by \(P\) as a result of the impulse.

1. A particle \(P\) of mass 0.5 kg is moving with velocity \(4 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse I Ns. Immediately after \(P\) receives the impulse, the velocity of \(P\) is \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).

Find

1. the magnitude of \(\mathbf { I }\),
2. the angle between \(\mathbf { I }\) and \(\mathbf { j }\).

2. \begin{figure}[h] \end{figure} The points \(A , B\) and \(C\) lie on a smooth horizontal plane. A small ball of mass 0.2 kg is moving along the line \(A B\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the ball is at \(B\), the ball is given an impulse. Immediately after the impulse is given, the ball moves along the line \(B C\) with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The line \(B C\) makes an angle of \(35 ^ { \circ }\) with the line \(A B\), as shown in Figure 1.

1. Find the magnitude of the impulse given to the ball.
2. Find the size of the angle between the direction of the impulse and the original direction of motion of the ball.

6. At time \(t\) seconds the acceleration, a \(\mathrm { m } \mathrm { s } ^ { - 2 }\), of a particle \(P\) relative to a fixed origin \(O\), is given by \(\mathbf { a } = 2 \mathbf { i } + 6 t \mathbf { j }\). Initially the velocity of \(P\) is \(( 2 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the velocity of \(P\) at time \(t\) seconds. At time \(t = 2\) seconds the particle \(P\) is given an impulse ( \(3 \mathbf { i } - 1.5 \mathbf { j }\) ) Ns. Given that the particle \(P\) has mass 0.5 kg ,
2. find the speed of \(P\) immediately after the impulse has been applied.

1. The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane. A ball of mass 0.5 kg is moving with velocity \(- 20 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is struck by a bat. The bat gives the ball an impulse of \(( 15 \mathbf { i } + 10 \mathbf { j } )\) Ns.

Find, to 3 significant figures, the speed of the ball immediately after it has been struck.
(5)

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string of natural length 2 m and modulus of elasticity 3 N . The other end of the string is attached to a fixed point \(O\) on a rough plane. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 2 } { 7 }\) The coefficient of friction between \(P\) and the plane is \(\frac { \sqrt { 5 } } { 5 }\) The particle \(P\) is initially at rest at the point \(O\), as shown in Figure 8. The particle \(P\) then receives an impulse of magnitude 4 Ns, directed up a line of greatest slope of the plane. The particle \(P\) moves up the plane and comes to rest at the point \(A\).

1. Find the extension of the elastic string when \(P\) is at \(A\).
2. Show that the particle does not remain at rest at \(A\).

3. A particle \(P\) of mass 0.5 kg moves in a straight line with simple harmonic motion, completing 4 oscillations per second. The particle comes to instantaneous rest at the fixed points \(A\) and \(B\), where \(A B = 0.5 \mathrm {~m}\).

1. Find the maximum magnitude of the acceleration of \(P\). When \(P\) is moving at its maximum speed it receives an impulse. The direction of this impulse is opposite to the direction in which \(P\) is moving when it receives the impulse. The impulse causes \(P\) to reverse its direction of motion but \(P\) continues to move with simple harmonic motion. The centre and period of this new simple harmonic motion are the same as the centre and period of the original simple harmonic motion. The amplitude is now half the original amplitude.
2. Find the magnitude of the impulse.
   \section*{II} " ; O L

7. A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic spring. The other end of the spring is attached to a fixed point \(O\) on a smooth horizontal table. The spring has natural length 2 m and modulus of elasticity 21.6 N . The particle \(P\) is placed on the table at the point \(A\), where \(O A = 2 \mathrm {~m}\). The particle \(P\) is now pulled away from \(O\) to the point \(B\), where \(O A B\) is a straight line with \(O B = 3.5 \mathrm {~m}\). It is then released from rest.

1. Prove that \(P\) moves with simple harmonic motion of period \(\frac { \pi } { 3 } \mathrm {~s}\).
2. Find the speed of \(P\) when it reaches \(A\). The point \(C\) is the mid-point of \(A B\).
3. Find, in terms of \(\pi\), the time taken for \(P\) to reach \(C\) for the first time. Later in the motion, \(P\) collides with a particle \(Q\) of mass 0.2 kg which is at rest at \(A\).
   After the impact, \(P\) and \(Q\) coalesce to form a single particle \(R\).
4. Show that \(R\) also moves with simple harmonic motion and find the amplitude of this motion. END

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.

7. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 1.2 m and modulus of elasticity 15 N . The other end of the spring is attached to a fixed point \(A\) on a smooth horizontal table. The particle is placed on the table at the point \(B\) where \(A B = 1.2 \mathrm {~m}\). The particle is pulled away from \(B\) to the point \(C\), where \(A B C\) is a straight line and \(B C = 0.8 \mathrm {~m}\), and is then released from rest.

1. 1. Show that \(P\) moves with simple harmonic motion with centre \(B\).
   2. Find the period of this motion.
2. Find the speed of \(P\) when it reaches \(B\). The point \(D\) is the midpoint of \(A B\).
3. Find the time taken for \(P\) to move directly from \(C\) to \(D\). When \(P\) first comes to instantaneous rest a particle \(Q\) of mass 0.3 kg is placed at \(B\). When \(P\) reaches \(B\) again, \(P\) strikes and adheres to \(Q\) to form a single particle \(R\).
4. Show that \(R\) also moves with simple harmonic motion.
5. Find the amplitude of this motion.
   

4 A small ball \(P\), of mass 40 grams, is dropped from rest at a point \(A\) which is 10 m above a fixed horizontal plane. At the same instant an identical ball \(Q\) is dropped from rest at the point \(B\), which is vertically below \(A\) and at a height of 5 m above the plane. The coefficient of restitution between \(Q\) and the plane is \(\frac { 1 } { 2 }\). Find the magnitude of the impulse exerted on \(Q\) by the plane. The balls collide after \(Q\) rebounds from the plane and before \(Q\) hits the plane again. Find the height above the plane of the point at which the collision occurs.

1 A bullet of mass \(m \mathrm {~kg}\) is fired into a fixed vertical barrier. It enters the barrier horizontally with speed \(280 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and emerges horizontally after 0.01 s with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a constant horizontal resisting force of magnitude 1500 N . Find \(m\).

1 A small smooth sphere \(P\) of mass \(2 m\) is at rest on a smooth horizontal surface. A horizontal impulse of magnitude \(8 m u\) is given to \(P\). Subsequently \(P\) collides directly with a fixed smooth vertical barrier at right angles to \(P\) 's direction of motion. Given that the coefficient of restitution between \(P\) and the barrier is 0.75 , find the speed of \(P\) after the collision.

1 Two uniform small smooth spheres, \(A\) and \(B\), of equal radii and masses 2 kg and 3 kg respectively, are at rest and not in contact on a smooth horizontal plane. Sphere \(A\) receives an impulse of magnitude 8 N s in the direction \(A B\). The coefficient of restitution between the spheres is \(e\). Find, in terms of \(e\), the speeds of \(A\) and \(B\) after \(A\) collides with \(B\). Given that the spheres move in opposite directions after the collision, show that \(e > \frac { 2 } { 3 }\).

1 A bullet of mass 0.01 kg is fired horizontally into a fixed vertical barrier which exerts a constant resisting force of magnitude 1000 N . The bullet enters the barrier with speed \(320 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and emerges with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). You may assume that the motion takes place in a horizontal straight line. Find

1. the magnitude of the impulse that acts on the bullet,
2. the thickness of the barrier,
3. the time taken for the bullet to pass through the barrier.

1 A bullet of mass \(m \mathrm {~kg}\) is fired horizontally into a fixed vertical block of material. It enters the block horizontally with speed \(250 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and emerges horizontally with speed \(70 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after 0.04 s . The block offers a constant horizontal resisting force of magnitude 450 N . Find the value of \(m\).

1 A bullet of mass 0.2 kg is fired into a fixed vertical barrier. It enters the barrier horizontally with speed \(250 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and emerges horizontally after a time \(T\) seconds with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a constant horizontal resisting force of magnitude 1200 N . Find \(T\).

4 Two smooth spheres \(A\) and \(B\), of equal radii, have masses 0.1 kg and \(m \mathrm {~kg}\) respectively. They are moving towards each other in a straight line on a smooth horizontal table and collide directly. Immediately before collision the speed of \(A\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Assume that in the collision \(A\) does not change direction. The speeds of \(A\) and \(B\) after the collision are \(v _ { A } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(v _ { B } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Express \(m\) in terms of \(v _ { A }\) and \(v _ { B }\), and hence show that \(m < 0.25\).
2. Assume instead that \(m = 0.2\) and that the coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\). Find the magnitude of the impulse acting on \(A\) in the collision.

4 A particle \(P\) of mass \(2 m\), moving on a smooth horizontal plane with speed \(u\), strikes a fixed smooth vertical barrier. Immediately before the collision the angle between the direction of motion of \(P\) and the barrier is \(60 ^ { \circ }\). The coefficient of restitution between \(P\) and the barrier is \(\frac { 1 } { 3 }\). Show that \(P\) loses two-thirds of its kinetic energy in the collision. Subsequently \(P\) collides directly with a particle \(Q\) of mass \(m\) which is moving on the plane with speed \(u\) towards \(P\). The magnitude of the impulse acting on each particle in the collision is \(\frac { 2 } { 3 } m u ( 1 + \sqrt { 3 } )\).

1. Show that the speed of \(P\) after this collision is \(\frac { 1 } { 3 } u\).
2. Find the exact value of the coefficient of restitution between \(P\) and \(Q\).

3. A particle of mass 0.6 kg is moving with constant velocity ( \(c \mathbf { i } + 2 c \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\), where \(c\) is a positive constant. The particle receives an impulse of magnitude \(2 \sqrt { 10 } \mathrm {~N} \mathrm {~s}\). Immediately after receiving the impulse the particle has velocity ( \(2 c \mathbf { i } - c \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\). Find the value of \(c\).
(6)

7. Two particles \(A\) and \(B\), of mass \(2 m\) and \(3 m\) respectively, are initially at rest on a smooth horizontal surface. Particle \(A\) is projected with speed \(3 u\) towards \(B\). Particle \(A\) collides directly with particle \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 3 } { 4 }\)

1. Find
   1. the speed of \(A\) immediately after the collision,
   2. the speed of \(B\) immediately after the collision. After the collision \(B\) hits a fixed smooth vertical wall and rebounds. The wall is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(e\). The magnitude of the impulse received by \(B\) when it hits the wall is \(\frac { 27 } { 4 } m u\).
2. Find the value of \(e\).
3. Determine whether there is a further collision between \(A\) and \(B\) after \(B\) rebounds from the wall.
   

1 A particle \(P\) is projected vertically downwards with initial speed \(3.5 \mathrm {~ms} ^ { - 1 }\) from a point \(A\) which is 5 m above horizontal ground.

1. Find the speed of \(P\) immediately before it strikes the ground. After striking the ground, \(P\) rebounds and moves vertically upwards and 0.87 s after leaving the ground \(P\) passes through \(A\).
2. Calculate the speed of \(P\) immediately after it leaves the ground. It is given that the mass of \(P\) is 0.2 kg .
3. Calculate the change in the momentum of \(P\) as a result of its collision with the ground.

3 A small sphere of mass 0.2 kg is projected vertically downwards with speed \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point at a height of 40 m above horizontal ground. It hits the ground and rebounds vertically upwards, coming to instantaneous rest at its initial point of projection. Ignoring air resistance, calculate

1. the coefficient of restitution between the sphere and the ground,
2. the magnitude of the impulse which the ground exerts on the sphere.