Collision with coalescing particles

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\includegraphics[max width=\textwidth, alt={}, center]{1b542910-a57e-4f58-a19f-92e67ee9bdf7-08_323_515_260_813} 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 is held with the string taut and horizontal. It is projected downwards with speed \(\sqrt { } ( 12 a g )\). At the lowest point of its motion, \(P\) collides directly with a particle \(Q\) of mass \(k m\) which is at rest (see diagram). In the collision, \(P\) and \(Q\) coalesce. The tension in the string immediately after the collision is half of its value immediately before the collision. Find the possible values of \(k\).

A particle \(P\), of mass \(m\), is able to move in a vertical circle on the smooth inner surface of a sphere with centre \(O\) and radius \(a\). Points \(A\) and \(B\) are on the inner surface of the sphere and \(A O B\) is a horizontal diameter. Initially, \(P\) is projected vertically downwards with speed \(\sqrt { } \left( \frac { 21 } { 2 } a g \right)\) from \(A\) and begins to move in a vertical circle. At the lowest point of its path, vertically below \(O\), the particle \(P\) collides with a stationary particle \(Q\), of mass \(4 m\), and rebounds. The speed acquired by \(Q\), as a result of the collision, is just sufficient for it to reach the point \(B\).

1. Find the speed of \(P\) and the speed of \(Q\) immediately after their collision.
   In its subsequent motion, \(P\) loses contact with the inner surface of the sphere at the point \(D\), where the angle between \(O D\) and the upward vertical through \(O\) is \(\theta\).
2. Find \(\cos \theta\).

4 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 is held vertically above \(O\) with the string taut and then projected horizontally with speed \(\sqrt { } \left( \frac { 13 } { 3 } a g \right)\). It begins to move in a vertical circle with centre \(O\). When \(P\) is at its lowest point, it collides with a stationary particle of mass \(\lambda m\). The two particles coalesce.

1. Show that the speed of the combined particle immediately after the impact is \(\frac { 5 } { \lambda + 1 } \sqrt { } \left( \frac { 1 } { 3 } a g \right)\). In the subsequent motion, the string becomes slack when the combined particle is at a height of \(\frac { 1 } { 3 } a\) above the level of \(O\).
2. Find the value of \(\lambda\).
3. Find, in terms of \(m\) and \(g\), the instantaneous change in the tension in the string as a result of the collision.

4 A particle \(P\), of mass \(m \mathrm {~kg}\), collides with a particle \(Q\), of mass 2 kg Immediately before the collision the velocity of \(P\) is \(\left[ \begin{array} { c } 4 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(\left[ \begin{array} { c } - 3 \\ 5 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) As a result of the collision the particles coalesce into a single particle which moves with velocity \(\left[ \begin{array} { l } k \\ 0 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant. Find the value of \(k\).

3 Two particles, \(A\) and \(B\), are moving on a smooth horizontal plane when they collide. The mass of \(A\) is 6 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Before the collision, the velocity of \(A\) is \(\left[ \begin{array} { l } 2 \\ 4 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { r } 3 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). After the collision, the velocity of \(A\) is \(\left[ \begin{array} { l } 1 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { l } 7 \\ b \end{array} \right] \mathrm { ms } ^ { - 1 }\).

1. Find \(m\).
2. \(\quad\) Find \(b\).
   (2 marks)
   .......... \(\_\_\_\_\)
   \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-07_40_118_529_159}
   \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-07_39_117_623_159}

\begin{enumerate} \item A small ball \(A\) is moving with velocity \(( 7 \mathbf { i } + 12 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). It collides in mid-air with another ball \(B\), of mass 0.4 kg , moving with velocity \(( - \mathrm { i } + 7 \mathrm { j } ) \mathrm { ms } ^ { - 1 }\). Immediately after the collision, \(A\) has velocity \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and \(B\) has velocity \(( 6 \cdot 5 \mathbf { i } + 13 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
Calculate the mass of \(A\). \item A stick of mass 0.75 kg is at rest with one end \(X\) on a rough horizontal floor and the other end \(Y\) leaning against a smooth vertical wall. The coefficient of friction between the stick
\includegraphics[max width=\textwidth, alt={}, center]{46695667-272a-4ba3-9f48-1d21280aa4d2-1_202_492_690_1471}
and the floor is 0.6 . Modelling the stick as a uniform rod, find the smallest angle that the stick can make with the floor before it starts to slip. \item An engine of mass 20000 kg climbs a hill inclined at \(10 ^ { \circ }\) to the horizontal. The total nongravitational resistance to its motion has magnitude 35000 N and the maximum speed of the engine on the hill is \(15 \mathrm {~ms} ^ { - 1 }\).

1. Find, in kW , the maximum rate at which the engine can work.
2. Find the maximum speed of the engine when it is travelling on a horizontal track against the same non-gravitational resistance as before. \item Relative to a fixed origin \(O\), the points \(X\) and \(Y\) have position vectors \(( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m }\) and \(( 12 \mathbf { i } + \mathbf { j } )\) m respectively, where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in the directions due east and due north respectively. A particle \(P\) starts from \(X\), and \(t\) seconds later its position vector relative to \(O\) is \(( 2 t + 4 ) \mathbf { i } + \left( k t ^ { 2 } - 5 \right) \mathbf { j }\).

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1. A battering-ram consists of a wooden beam fixed to a trolley. The battering-ram runs along horizontal ground and collides directly with a vertical wall, as shown in Fig. 1.1. The batteringram has a mass of 4000 kg . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b2962c91-4739-4d1e-98f3-62d420f6dddf-2_310_793_424_717} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} Initially the battering-ram is at rest. Some men push it for 8 seconds and let go just as it is about to hit the wall. While the battering-ram is being pushed, the constant overall force on it in the direction of its motion is 1500 N .
   1. At what speed does the battering-ram hit the wall? The battering-ram hits a loose stone block of mass 500 kg in the wall. Linear momentum is conserved and the coefficient of restitution in the impact is 0.2 .
   2. Calculate the speeds of the stone block and of the battering-ram immediately after the impact.
   3. Calculate the energy lost in the impact.
2. Small objects A and B are sliding on smooth, horizontal ice. Object A has mass 4 kg and speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the \(\mathbf { i }\) direction. B has mass 8 kg and speed \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction shown in Fig. 1.2, where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b2962c91-4739-4d1e-98f3-62d420f6dddf-2_515_783_1637_721} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure}
   1. Write down the linear momentum of A and show that the linear momentum of B is \(( 36 \mathbf { i } + 36 \sqrt { 3 } \mathbf { j } )\) Ns. After the objects meet they stick together (coalesce) and move with a common velocity of \(( u \mathbf { i } + v \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
   2. Calculate \(u\) and \(v\).
   3. Find the angle between the direction of motion of the combined object and the \(\mathbf { i }\) direction. Make your method clear.

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1. In this part-question, all the objects move along the same straight line on a smooth horizontal plane. All their collisions are direct. The masses of the objects \(\mathrm { P } , \mathrm { Q }\) and R and the initial velocities of P and Q (but not R ) are shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-2_177_1011_488_529} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} A force of 21 N acts on P for 2 seconds in the direction \(\mathrm { PQ } . \mathrm { P }\) does not reach Q in this time.
   1. Calculate the speed of P after the 2 seconds. The force of 21 N is removed after the 2 seconds. When P collides with Q they stick together (coalesce) to form an object S of mass 6 kg .
   2. Show that immediately after the collision S has a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards R . The collision between S and R is elastic with coefficient of restitution \(\frac { 1 } { 4 }\). After the collision, S has a velocity of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of its motion before the collision.
   3. Find the velocities of R before and after the collision. \section*{(b) In this part-question take \(\boldsymbol { g } = \mathbf { 1 0 }\).} A particle of mass 0.2 kg is projected vertically downwards with initial speed \(5 \mathrm {~ms} ^ { - 1 }\) and it travels 10 m before colliding with a fixed smooth plane. The plane is inclined at \(\alpha\) to the vertical where \(\tan \alpha = \frac { 3 } { 4 }\). Immediately after its collision with the plane, the particle has a speed of \(13 \mathrm {~ms} ^ { - 1 }\). This information is shown in Fig. 1.2. Air resistance is negligible. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-2_383_341_1795_854} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure}
   4. Calculate the angle between the direction of motion of the particle and the plane immediately after the collision. Calculate also the coefficient of restitution in the collision.
   5. Calculate the magnitude of the impulse of the plane on the particle.

5 A particle \(P\), of mass 2.5 kg , is in equilibrium suspended from a fixed point \(A\) by a light elastic string of natural length 3 m and modulus of elasticity 36.75 N . Another particle \(Q\), of mass 1 kg , is released from rest at \(A\) and falls freely until it reaches \(P\) and becomes attached to it.

1. Show that the speed of the combined particles, immediately after \(Q\) becomes attached to \(P\), is \(2 \sqrt { 2 } \mathrm {~ms} ^ { - 1 }\). The combined particles fall a further distance \(X \mathrm {~m}\) before coming to instantaneous rest.
2. Find a quadratic equation satisfied by \(X\), and show that it simplifies to \(35 X ^ { 2 } - 56 X - 80 = 0\).