Collision followed by wall impact

7. A smooth sphere \(A\) of mass \(4 m\) is moving on a smooth horizontal plane with speed \(u\). It collides directly with a stationary smooth sphere \(B\) of mass \(5 m\) and with the same radius as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 2 }\).

1. Show that after the collision the speed of \(B\) is 4 times greater than the speed of \(A\).
   (7 marks)
   Sphere \(B\) subsequently hits a smooth vertical wall at right angles. After rebounding from the wall, \(B\) collides with \(A\) again and as a result of this collision, \(B\) comes to rest. Given that the coefficient of restitution between \(B\) and the wall is \(e\),
2. find \(e\). END

4 Two small smooth spheres, \(A\) and \(B\), of equal radii have masses 0.3 kg and 0.2 kg respectively. They are moving on a smooth horizontal surface directly towards each other with speeds \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively when they collide. The coefficient of restitution between \(A\) and \(B\) is 0.8 .

1. Find the speeds of \(A\) and \(B\) immediately after the collision.
2. Subsequently, \(B\) collides with a fixed smooth vertical wall which is at right angles to the path of the sphere. The coefficient of restitution between \(B\) and the wall is 0.7 . Show that \(B\) will collide again with \(A\).

3 The diagram shows two blocks P and Q of masses 0.5 kg and 2 kg respectively, on a horizontal surface. The points \(\mathrm { A } , \mathrm { B }\) and C lie on the surface in a straight line. There is a wall at C . The surface between B and C is smooth, and the surface between A and B is rough, such that the coefficient of friction between P and AB is \(\frac { 2 } { 3 }\). \includegraphics[max width=\textwidth, alt={}, center]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-3_229_1271_1601_278} P is projected with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) directly towards Q , which is at rest. As a result of the collision between P and Q, P changes direction and subsequently comes to rest at A. You may assume that P only collides with Q once.

1. Determine the coefficient of restitution between P and Q .
2. Calculate the impulse exerted on P by Q during their collision. After colliding with P , Q strikes the wall, which is perpendicular to the direction of the motion of Q , and comes to rest exactly halfway between A and B . The collision between Q and the wall is perfectly elastic.
3. Determine the coefficient of friction between Q and AB .

6 Stones A and B have masses \(m \mathrm {~kg}\) and \(3 m \mathrm {~kg}\) respectively. They lie at rest on a large area of smooth horizontal ice and may move freely over the ice. Stone A is given a horizontal impulse of magnitude \(m u \mathrm {~N} s\) towards B so that the stones collide directly. After the collision the direction of motion of A is reversed. The coefficient of restitution between A and B is denoted by \(e\).

1. Find the range of possible values of \(e\). After the collision, B subsequently collides with a vertical smooth wall perpendicular to its path and rebounds. The coefficient of restitution between \(B\) and the wall is the same as the coefficient of restitution between A and B .
2. Show that A and B will collide again unless the collision between B and the wall is perfectly elastic.
3. Explain why modelling the collision between B and the wall as perfectly elastic is possibly unrealistic.
4. Given that the kinetic energy lost in the first collision between A and B is \(\frac { 5 } { 24 } m u ^ { 2 }\), determine the value of \(e\).
5. Given that B was 2 metres from the wall when the stones first collided, determine the distance of the stones from the wall when they next collide.

6 A block rests on a horizontal surface. The coefficient of friction between the block and the surface is \(\mu\).

1. Show that if the block is given an initial speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it will move a distance of \(\frac { \mathrm { v } ^ { 2 } } { 2 \mu \mathrm {~g} }\) before coming to rest. Block B rests on the same horizontal surface as a sphere S . On the other side of S is a vertical wall, as shown below. The mass of \(B\) is 8 times the mass of \(S\). \includegraphics[max width=\textwidth, alt={}, center]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-8_211_1013_662_244} S is projected directly towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and hits B . It is given that
   Furthermore, you should model the contact between B and the surface as rough and model the contact between S and the surface as smooth.
2. Determine, in terms of \(u\), expressions for
   It is given that B has sufficient time to come to rest before each subsequent collision with S .
   Let \(\mathrm { X } _ { \mathrm { n } }\) be the distance B moves after the \(n\)th impact between S and B .
3. Explain why \(\mathrm { x } _ { \mathrm { n } + 1 } = \frac { 9 } { 25 } \mathrm { x } _ { \mathrm { n } }\).
4. Given that \(u = 11.2\) and the coefficient of friction between B and the surface is \(\frac { 1 } { 7 }\), show that B will travel a total distance that cannot exceed 2.8 m . \section*{END OF QUESTION PAPER} \section*{OCR
   Oxford Cambridge and RSA}

1. A particle \(P\) of mass \(3 m\) is moving in a straight line on a smooth horizontal floor. A particle \(Q\) of mass \(5 m\) is moving in the opposite direction to \(P\) along the same straight line.

The particles collide directly.
Immediately before the collision, the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(u\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Show that the speed of \(Q\) immediately after the collision is \(\frac { u } { 8 } ( 9 e + 1 )\)
2. Find the range of values of \(e\) for which the direction of motion of \(P\) is not changed as a result of the collision. When \(P\) and \(Q\) collide they are at a distance \(d\) from a smooth fixed vertical wall, which is perpendicular to their direction of motion. After the collision with \(P\), particle \(Q\) collides directly with the wall and rebounds so that there is a second collision between \(P\) and \(Q\). This second collision takes place at a distance \(x\) from the wall. Given that \(e = \frac { 1 } { 18 }\) and the coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\)
3. find \(x\) in terms of \(d\).

A particle \(P\) of mass \(2 m \mathrm {~kg}\) is moving with speed \(2 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal plane. Particle \(P\) collides with a particle \(Q\) of mass \(3 m \mathrm {~kg}\) which is at rest on the plane. The coefficient of restitution between \(P\) and \(Q\) is \(e\). Immediately after the collision the speed of \(Q\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. Show that \(v = \frac { 4 u ( 1 + e ) } { 5 }\)
2. Show that \(\frac { 4 u } { 5 } \leqslant v \leqslant \frac { 8 u } { 5 }\) Given that the direction of motion of \(P\) is reversed by the collision,
3. find, in terms of \(u\) and \(e\), the speed of \(P\) immediately after the collision. After the collision, \(Q\) hits a wall, that is fixed at right angles to the direction of motion of \(Q\), and rebounds. The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 6 }\) Given that \(P\) and \(Q\) collide again,
4. find the full range of possible values of \(e\).

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) and a particle \(Q\) of mass \(4 m\) are at rest on a smooth horizontal plane, as shown in Figure 2. Particle \(P\) is projected with speed \(u\) along the plane towards \(Q\) and the particles collide.
The coefficient of restitution between the particles is \(e\), where \(e > \frac { 1 } { 4 }\) As a result of the collision, the direction of motion of \(P\) is reversed and \(P\) has speed \(\frac { u } { 5 } ( 4 e - 1 )\).

1. Find, in terms of \(u\) and \(e\), the speed of \(Q\) after the collision. After the collision, \(P\) goes on to hit a vertical wall which is fixed at right angles to the direction of motion of \(P\). The coefficient of restitution between \(P\) and the wall is \(f\), where \(f > 0\) Given that \(e = \frac { 3 } { 4 }\)
2. find, in terms of \(m , u\) and \(f\), the kinetic energy lost by \(P\) as a result of its impact with the wall. Give your answer in its simplest form. After its impact with the wall, \(P\) goes on to collide with \(Q\) again.
3. Find the complete range of possible values of \(f\).

A particle P of mass 3 m is moving in a straight line on a smooth horizontal table. A particle \(Q\) of mass \(m\) is moving in the opposite direction to \(P\) along the same straight line. The particles collide directly. Immediately before the collision the speed of P is u and the speed of Q is 2 u . The velocities of P and Q immediately after the collision, measured in the direction of motion of P before the collision, are V and W respectively. The coefficient of restitution between P and Q is e .

1. Find an expression for v in terms of u and e . Given that the direction of motion of P is changed by the collision,
2. find the range of possible values of e.
3. Show that \(\mathrm { w } = \frac { \mathrm { u } } { 4 } ( 1 + 9 \mathrm { e } )\). Following the collision with P , the particle Q then collides with and rebounds from a fixed vertical wall which is perpendicular to the direction of motion of \(Q\). The coefficient of restitution between \(Q\) and the wall is \(f\).
   Given that \(\mathrm { e } = \frac { 5 } { 9 }\), and that P and Q collide again in the subsequent motion,
4. find the range of possible values of f .

   VIIIV SIHI NI JIIIM ION OC

   VIIIV SIHI NI JIHM I I ON OC

   VIAV SIHI NI JIIIM I ON OC

   \section*{Q uestion 4 continued}

Two particles, \(P\) and \(Q\), are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.
The mass of \(P\) is \(3 m\) and the mass of \(Q\) is \(4 m\).
Immediately before the collision the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(u\).
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Show that the speed of \(Q\) immediately after the collision is \(\frac { u } { 7 } ( 9 e + 2 )\) After the collision with \(P\), particle \(Q\) collides directly with a fixed vertical wall and rebounds. The wall is perpendicular to the direction of motion of \(Q\).
   The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 2 }\)
2. Find the complete range of possible values of \(e\) for which there is a second collision between \(P\) and \(Q\).

A particle \(P\) of mass \(2 m\) is moving in a straight line with speed \(3 u\) on a smooth horizontal plane. It collides directly with a particle \(Q\) of mass \(m\) that is moving on the plane with speed \(2 u\) in the opposite direction to \(P\).
The coefficient of restitution between \(P\) and \(Q\) is \(e\), where \(e > \frac { 4 } { 5 }\)

1. Show that the speed of \(Q\) immediately after the collision is \(\frac { ( 4 + 10 e ) u } { 3 }\) After the collision \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). The coefficient of restitution between \(Q\) and the wall is \(f\).
2. Find, in terms of \(\boldsymbol { e }\), the set of values of \(f\) for which there will be a second collision between \(P\) and \(Q\).

1. A particle \(A\) of mass \(2 m\) is moving in a straight line with speed \(3 u\) on a smooth horizontal plane. Particle \(A\) collides directly with a particle \(B\) of mass \(m\) which is at rest on the plane.

The coefficient of restitution between \(A\) and \(B\) is \(e\), where \(e > 0\)

1. Show that the speed of \(B\) immediately after the collision is \(2 u ( 1 + e )\). After the collision, \(B\) hits a smooth fixed vertical wall which is perpendicular to the direction of motion of \(B\).
2. Show that there will be a second collision between \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\) Find, in simplified form, in terms of \(m\), \(u\) and \(e\),
3. the magnitude of the impulse received by \(B\) in its collision with the wall,
4. the loss in kinetic energy of \(B\) due to its collision with the wall.

5 Two particles \(A\) and \(B\) are on a smooth horizontal floor with \(B\) between \(A\) and a vertical wall. The masses of \(A\) and \(B\) are 4 kg and 11 kg respectively. Initially, \(B\) is at rest and \(A\) is moving towards \(B\) with a speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). \(A\) collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(e\). \includegraphics[max width=\textwidth, alt={}, center]{bf86ac88-0fd1-4d49-a705-9b8d06fbac2a-3_209_803_1658_630}

1. Show that immediately after the collision the speed of \(B\) is \(\frac { 4 } { 15 } u ( 1 + e )\). After the collision between \(A\) and \(B\) the direction of motion of \(A\) is reversed. \(B\) subsequently collides directly with the vertical wall. The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 } e\).
2. Given that there is a second collision between \(A\) and \(B\), find the range of possible values of \(e\).

6 A smooth sphere \(A\) of mass \(m\) is moving with speed \(5 u\) in a straight line on a smooth horizontal table. The sphere \(A\) collides directly with a smooth sphere \(B\) of mass \(7 m\), having the same radius as \(A\) and moving with speed \(u\) in the same direction as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(e\). \includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-5_287_880_529_571}

1. Show that the speed of \(B\) after the collision is \(\frac { u } { 2 } ( e + 3 )\).
2. Given that the direction of motion of \(A\) is reversed by the collision, show that \(e > \frac { 3 } { 7 }\).
3. Subsequently, \(B\) hits a wall fixed at right angles to the direction of motion of \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\). Given that after \(B\) rebounds from the wall both spheres move in the same direction and collide again, show also that \(e < \frac { 9 } { 13 }\).
   (4 marks)

1. A particle \(P\) has mass \(5 m\) and a particle \(Q\) has mass \(2 m\).

The particles are moving in opposite directions along the same straight line on a smooth horizontal surface.
Particle \(P\) collides directly with particle \(Q\).
Immediately before the collision, the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(3 u\). Immediately after the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\).
The direction of motion of \(Q\) is reversed as a result of the collision.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the set of values of \(e\) for which the direction of motion of \(P\) is unchanged as a result of the collision. In the collision, \(Q\) receives an impulse of magnitude \(\frac { 60 } { 7 } m u\)
2. Show that \(e = \frac { 1 } { 5 }\) After the collision, \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). Particle \(Q\) rebounds and there is a second collision between \(P\) and \(Q\).
   The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\)
3. Find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(Q\) in the second collision between \(P\) and \(Q\).

Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(3m\) and \(m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is at rest. The coefficient of restitution between the spheres is \(e\).

1. Find, in terms of \(u\) and \(e\), expressions for the velocities of \(A\) and \(B\) after the collision. [3]

Sphere \(B\) continues to move until it strikes a fixed smooth vertical barrier which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the barrier is \(\frac{3}{4}\). When the spheres subsequently collide, \(A\) is brought to rest.

1. Find the value of \(e\). [7]

Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(3m\) and \(m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is at rest. The coefficient of restitution between the spheres is \(e\).

1. Find, in terms of \(u\) and \(e\), expressions for the velocities of \(A\) and \(B\) after the collision. [3]
2. Sphere \(B\) continues to move until it strikes a fixed smooth vertical barrier which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the barrier is \(\frac{3}{4}\). When the spheres subsequently collide, \(A\) is brought to rest. Find the value of \(e\). [7]

Two uniform small smooth spheres \(A\) and \(B\) have equal radii and each has mass \(m\). Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is at rest. The coefficient of restitution between the spheres is \(\frac{2}{3}\). Sphere \(B\) is initially at a distance \(d\) from a fixed smooth vertical wall which is perpendicular to the direction of motion of \(A\). The coefficient of restitution between \(B\) and the wall is \(\frac{1}{3}\).

1. Show that the speed of \(B\) after its collision with the wall is \(\frac{5}{18}u\). [4]
2. Find the distance of \(B\) from the wall when it collides with \(A\) for the second time. [6]

Two small smooth spheres \(A\) and \(B\) of equal radius have masses \(m\) and \(3m\) respectively. They lie at rest on a smooth horizontal plane with their line of centres perpendicular to a smooth fixed vertical barrier wall \(9\) feet away from the barrier. The coefficient of restitution between \(A\) and \(B\), and between \(B\) and the barrier, is \(e\), where \(e > \frac{1}{4}\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\). Show that after colliding with \(B\) the direction of motion of \(A\) is reversed. [5] After the impact, \(B\) hits the barrier and rebounds. Show that \(B\) will subsequently collide with \(A\) again unless \(e = 1\). [3]

A small uniform sphere \(A\), of mass \(2m\), is moving with speed \(u\) on a smooth horizontal surface when it collides directly with a small uniform sphere \(B\), of mass \(m\), which is at rest. The spheres have equal radii and the coefficient of restitution between them is \(e\). Find expressions for the speeds of \(A\) and \(B\) immediately after the collision. [4] Subsequently \(B\) collides with a vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(0.4\). After \(B\) has collided with the wall, the speeds of \(A\) and \(B\) are equal. Find \(e\). [2] Initially \(B\) is at a distance \(d\) from the wall. Find the distance of \(B\) from the wall when it next collides with \(A\). [4]

Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(2m\) and \(m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is at rest. The coefficient of restitution between the spheres is \(\frac{2}{3}\).

1. Find, in terms of \(u\), the speeds of \(A\) and \(B\) after this collision. [4]
2. Sphere \(B\) is initially at a distance \(d\) from a fixed smooth vertical wall which is perpendicular to the direction of motion of \(A\). The coefficient of restitution between \(B\) and the wall is \(\frac{1}{2}\). Find, in terms of \(d\) and \(u\), the time that elapses between the first and second collisions between \(A\) and \(B\). [5]

\includegraphics{figure_2} A smooth sphere \(P\) of mass \(2m\) is moving in a straight line with speed \(v\) on a smooth horizontal table. Another smooth sphere \(Q\) of mass \(m\) is at rest on the table. The sphere \(P\) collides directly with \(Q\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{3}\). The spheres are modelled as particles.

1. Show that, immediately after the collision, the speeds of \(P\) and \(Q\) are \(\frac{3}{4}u\) and \(\frac{5}{4}u\) respectively. [7]

After the collision, \(Q\) strikes a fixed vertical wall which is perpendicular to the direction of motion of \(P\) and \(Q\). The coefficient of restitution between \(Q\) and the wall is \(e\). When \(P\) and \(Q\) collide again, \(P\) is brought to rest.

1. Find the value of \(e\). [7]

1. Explain why there must be a third collision between \(P\) and \(Q\). [1]

Show that \(GX = \frac{44}{63}a\). [6] The mass of the lamina is \(M\). A particle of mass \(λM\) is attached to the lamina at \(C\). The lamina is suspended from \(B\) and hangs freely under gravity with \(AB\) horizontal.

1. Find the value of \(λ\). [3]

TURN OVER FOR QUESTION 7

A smooth sphere \(P\) of mass \(2m\) is moving in a straight line with speed \(u\) on a smooth horizontal table. Another smooth sphere \(Q\) of mass \(m\) is at rest on the table. The sphere \(P\) collides directly with \(Q\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{3}\). The spheres are modelled as particles.

1. Show that, immediately after the collision, the speeds of \(P\) and \(Q\) are \(\frac{2}{9}u\) and \(\frac{8}{9}u\) respectively. [7]

After the collision, \(Q\) strikes a fixed vertical wall which is perpendicular to the direction of motion of \(P\) and \(Q\). The coefficient of restitution between \(Q\) and the wall is \(e\). When \(P\) and \(Q\) collide again, \(P\) is brought to rest.

1. Find the value of \(e\). [7]
2. Explain why there must be a third collision between \(P\) and \(Q\). [1]

A particle \(A\) of mass \(2m\) is moving with speed \(3u\) in a straight line on a smooth horizontal table. The particle collides directly with a particle \(B\) of mass \(m\) moving with speed \(2u\) 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.

1. Calculate the coefficient of restitution between \(A\) and \(B\). [6]
2. Show that the kinetic energy lost in the collision is \(7mu^2\). [3]

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 \(\frac{14}{3}mu\).

1. Calculate the coefficient of restitution between \(B\) and the wall. [4]

A small smooth ball \(A\) of mass \(m\) is moving on a horizontal table with speed \(u\) when it collides directly with another small smooth ball \(B\) of mass \(3m\) which is at rest on the table. The balls have the same radius and the coefficient of restitution between the balls is \(e\). The direction of motion of \(A\) is reversed as a result of the collision.

1. Find, in terms of \(e\) and \(u\), the speeds of \(A\) and \(B\) immediately after the collision. [7]

In the subsequent motion \(B\) strikes a vertical wall, which is perpendicular to the direction of motion of \(B\), and rebounds. The coefficient of restitution between \(B\) and the wall is \(\frac{1}{3}\). Given that there is a second collision between \(A\) and \(B\),

1. find the range of values of \(e\) for which the motion described is possible. [6]