Collision with vector velocities

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)

2 A particle, \(A\), of mass 12 kg is moving on a smooth horizontal surface with velocity \(\left[ \begin{array} { l } 4 \\ 7 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). It then collides and coalesces with a second particle, \(B\), of mass 4 kg .

1. If before the collision the velocity of \(B\) was \(\left[ \begin{array} { l } 2 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), find the velocity of the combined particle after the collision.
2. If after the collision the velocity of the combined particle is \(\left[ \begin{array} { l } 1 \\ 4 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), find the velocity of \(B\) before the collision.

4 Two particles, \(A\) and \(B\), are moving on a horizontal plane when they collide and coalesce to form a single particle. The mass of \(A\) is 5 kg and the mass of \(B\) is 15 kg . Before the collision, the velocity of \(A\) is \(\left[ \begin{array} { c } 2 U \\ U \end{array} \right] \mathrm { ms } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { c } V \\ - 1 \end{array} \right] \mathrm { ms } ^ { - 1 }\). After the collision, the velocity of the combined particle is \(\left[ \begin{array} { l } V \\ 0 \end{array} \right] \mathrm { ms } ^ { - 1 }\).

1. Find:
   1. \(U\);
   2. \(V\).
2. Find the speed of \(A\) before the collision.

1 Two particles, \(A\) of mass 7 kg and \(B\) of mass 3 kg , are moving on a smooth horizontal plane when they collide. Just before the collision, the velocity of \(A\) is \(( 3 \mathbf { i } + 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( 6 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). During the collision, the particles coalesce to form a single combined particle. Find the velocity of the single combined particle after the collision.

1 A particle of mass \(m\) has velocity \(\left[ \begin{array} { l } 4 \\ 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). It then collides with a particle of mass 3 kg which has velocity \(\left[ \begin{array} { l } - 1 \\ - 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). During the collision the particles coalesce and move with velocity \(\left[ \begin{array} { l } 1 \\ V \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Show that \(m = 2\).
2. Find \(V\).

8 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface.
The particle \(A\) has mass \(m \mathrm {~kg}\) and is moving with velocity \(\left[ \begin{array} { r } 5 \\ - 3 \end{array} \right] \mathrm { ms } ^ { - 1 }\). The particle \(B\) has mass 0.2 kg and is moving with velocity \(\left[ \begin{array} { l } 2 \\ 3 \end{array} \right] \mathrm { ms } ^ { - 1 }\).

1. Find, in terms of \(m\), an expression for the total momentum of the particles.
2. The particles \(A\) and \(B\) collide and form a single particle \(C\), which moves with velocity \(\left[ \begin{array} { c } k \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant.
   1. Show that \(m = 0.1\).
   2. Find the value of \(k\).

1 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface when they collide. During the collision, the two particles coalesce to form a single combined particle. Particle \(A\) has mass 3 kg and particle \(B\) has mass 7 kg . Before the collision, the velocity of \(A\) is \(\left[ \begin{array} { r } 6 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { r } - 1 \\ 4 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the velocity of the combined particle after the collision.
2. Find the speed of the combined particle after the collision.

4 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface 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 \(( 5 \mathbf { i } + 18 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( 2 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). After the collision, the velocity of \(A\) is \(8 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) and the velocity of \(B\) is \(V \mathbf { j } \mathrm {~ms} ^ { - 1 }\).

1. Find \(m\).
2. \(\quad\) Find \(V\).

5 Two particles, \(A\) and \(B\), have masses of \(m\) and \(k m\) respectively, where \(k\) is a constant. The particles are moving on a smooth horizontal plane when they collide and coalesce to form a single particle. Just before the collision the velocities of \(A\) and \(B\) are \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and \(( 6 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) respectively. Immediately after the collision the combined particle has velocity \(( 5.2 \mathbf { i } - 0.4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find \(k\).
[0pt] [5 marks]

5. A cricket ball of mass 0.3 kg is approaching a batsman at \({ } ^ { - } 30 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The batsman hits the ball with a 1.5 kg bat moving with velocity \(15 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\). Contact between bat and ball lasts for 0.2 seconds. Immediately after this, bat and ball move with velocities \(5 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) and \(v \mathbf { i } \mathrm {~ms} ^ { - 1 }\) respectively.

1. Suggest a suitable model for the cricket ball.
2. Calculate the value of \(v\).
3. Find the magnitude of the force with which the batsman hits the ball.

1. A snooker ball \(A\) is moving on a horizontal table with velocity \(( 5 \mathbf { i } + 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).

It collides with another ball \(B\), whose mass is twice the mass of \(A\).
After the collision, \(A\) has velocity \(( - 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and \(B\) has velocity \(( \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
Find the velocity of \(B\) before the collision.

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.]

A particle \(A\) has mass 3 kg and a particle \(B\) has mass 2 kg .
The particles are moving on a smooth horizontal plane when they collide directly.
Immediately before the collision, the velocity of \(A\) is \(( 3 \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the velocity of \(B\) is \(( - 6 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) Immediately after the collision the velocity of \(A\) is \(\left( - 2 \mathbf { i } + \frac { 2 } { 3 } \mathbf { j } \right) \mathrm { ms } ^ { - 1 }\)

1. Find the total kinetic energy of the two particles before the collision.
2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the impulse exerted on \(A\) by \(B\) in the collision.
3. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) immediately after the collision.

7. \begin{figure}[h] \end{figure} A ball \(B\) of mass 0.4 kg is struck by a bat at a point \(O\) which is 1.2 m above horizontal ground. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are respectively horizontal and vertical. Immediately before being struck, \(B\) has velocity \(( - 20 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after being struck it has velocity \(( 15 \mathbf { i } + 16 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). After \(B\) has been struck, it moves freely under gravity and strikes the ground at the point \(A\), as shown in Fig. 3. The ball is modelled as a particle.

1. Calculate the magnitude of the impulse exerted by the bat on \(B\).
2. By using the principle of conservation of energy, or otherwise, find the speed of \(B\) when it reaches \(A\).
3. Calculate the angle which the velocity of \(B\) makes with the ground when \(B\) reaches \(A\).
4. State two additional physical factors which could be taken into account in a refinement of the model of the situation which would make it more realistic.