Velocity after impulse applied

1. A particle \(P\) of mass 1.5 kg is moving with velocity \(( 4 \mathbf { i } + 6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse of magnitude 15Ns. Immediately after \(P\) receives the impulse, the velocity of \(P\) is \(\boldsymbol { v } \mathrm { m } \mathrm { s } ^ { - 1 }\).

Find the two possible values of \(v\).

5. A particle of mass 0.5 kg is moving on a smooth horizontal surface with velocity \(12 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse \(K ( \mathbf { i } + \mathbf { j } ) \mathrm { N } \mathrm { s }\), where \(K\) is a positive constant. Immediately after receiving the impulse the particle is moving with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction which makes an acute angle \(\theta\) with the vector \(\mathbf { i }\). Find

1. the value of \(K\),
2. the size of angle \(\theta\).

1. A particle of mass 4 kg is moving with velocity \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) when it receives an impulse of \(( 7 \mathbf { i } - 5 \mathbf { j } )\) Ns.

Find the speed of the particle immediately after receiving the impulse.

1. A particle of mass 2 kg is moving with velocity \(( 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) when it receives an impulse \(\mathbf { I N }\) s, such that \(\mathbf { I } = a \mathbf { i } + b \mathbf { j }\)

Immediately after receiving the impulse, the particle is moving with velocity \(\lambda ( \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(\lambda\) is a constant. Given that the magnitude of \(\mathbf { I }\) is \(\sqrt { 40 }\), find the two possible impulses.
(5)

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.3 kg is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight line on a smooth horizontal plane. The particle receives a horizontal impulse of magnitude \(J\) Ns. The speed of \(P\) immediately after receiving the impulse is \(8 \mathrm {~ms} ^ { - 1 }\). The angle between the direction of motion of \(P\) before it receives the impulse and the direction of the impulse is \(60 ^ { \circ }\), as shown in Figure 2. Find the value of \(J\).
(6)

3. A particle \(P\) of mass 0.5 kg is moving with velocity \(\lambda ( \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when \(P\) receives an impulse of magnitude \(\sqrt { \frac { 5 } { 2 } } \mathrm { Ns }\) Immediately after \(P\) receives the impulse, the velocity of \(P\) is \(4 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) Given that \(\lambda\) is a constant, find the two possible values of \(\lambda\)

2. A particle of mass 2 kg is moving with velocity \(3 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives an impulse \(( \lambda \mathbf { i } - 2 \lambda \mathbf { j } )\) Ns, where \(\lambda\) is a constant. Immediately after the impulse is received, the speed of the particle is \(6 \mathrm {~ms} ^ { - 1 }\). Find the possible values of \(\lambda\).

1. \begin{figure}[h] \end{figure} A particle, \(P\), of mass 0.8 kg , moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight line on a smooth horizontal plane, receives a horizontal impulse of magnitude 6 N s. The angle between the initial direction of motion of \(P\) and the direction of the impulse is \(50 ^ { \circ }\), as shown in Figure 1. Find the speed of \(P\) immediately after receiving the impulse.

4. A particle \(P\) of mass 0.75 kg is moving with velocity \(4 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives an impulse \(\mathbf { J }\) Ns. Immediately after \(P\) receives the impulse, the speed of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Given that \(\mathbf { J } = c ( - \mathbf { i } + 2 \mathbf { j } )\), where \(c\) is a constant, find the two possible values of \(c\).
(6)

1. A particle \(P\) of mass 0.2 kg is moving with velocity \(( 4 \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)

The particle receives an impulse \(\lambda ( \mathbf { i } + \mathbf { j } ) \mathrm { Ns }\), where \(\lambda\) is a constant.
Immediately after receiving the impulse, the speed of \(P\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
Find the possible values of \(\lambda\)

3. \begin{figure}[h] \end{figure} A particle \(Q\) of mass 0.25 kg is moving in a straight line on a smooth horizontal surface with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse of magnitude \(I \mathrm { Ns }\). The impulse acts parallel to the horizontal surface and at \(60 ^ { \circ }\) to the original direction of motion of \(Q\). Immediately after receiving the impulse, the speed of \(Q\) is \(12 \mathrm {~ms} ^ { - 1 }\)
As a result of receiving the impulse, the direction of motion of \(Q\) is turned through \(\alpha ^ { \circ }\), as shown in Figure 2. Find the value of \(I\)

2. A particle \(P\) of mass 0.75 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of \(P\) is given by $$\mathbf { v } = \left( t ^ { 2 } + 2 \right) \mathbf { i } - 6 t \mathbf { j }$$

1. Find the magnitude of \(\mathbf { F }\) when \(t = 4\).
   (5) When \(t = 5\), the particle \(P\) receives an impulse of magnitude \(9 \sqrt { } 2 \mathrm { Ns }\) in the direction of the vector \(\mathbf { i } - \mathbf { j }\).
2. Find the velocity of \(P\) immediately after the impulse.

1. A tennis ball of mass 0.1 kg is hit by a racquet. Immediately before being hit, the ball has velocity \(30 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The racquet exerts an impulse of \(( - 2 \mathbf { i } - 4 \mathbf { j } ) \mathrm { Ns }\) on the ball. By modelling the ball as a particle, find the velocity of the ball immediately after being hit.
2. A particle \(P\) is moving in a plane. At time \(t\) seconds, \(P\) is moving with velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(\mathbf { v } = 2 t \mathbf { i } - 3 t ^ { 2 } \mathbf { j }\).

Find

1. the speed of \(P\) when \(t = 4\)
2. the acceleration of \(P\) when \(t = 4\) Given that \(P\) is at the point with position vector \(( - 4 \mathbf { i } + \mathbf { j } ) \mathrm { m }\) when \(t = 1\),
3. find the position vector of \(P\) when \(t = 4\)

4. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) lie in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) vertical. A ball of mass 0.1 kg is hit by a bat which gives it an impulse of ( \(3.5 \mathbf { i } + 3 \mathbf { j }\) ) Ns. The velocity of the ball immediately after being hit is \(( 10 \mathbf { i } + 25 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the velocity of the ball immediately before it is hit. In the subsequent motion the ball is modelled as a particle moving freely under gravity. When it is hit the ball is 1 m above horizontal ground.
2. Find the greatest height of the ball above the ground in the subsequent motion. The ball is caught when it is again 1 m above the ground.
3. Find the distance from the point where the ball is hit to the point where it is caught.

1. A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds,

$$\mathbf { F } = ( 6 t - 5 ) \mathbf { i } + \left( t ^ { 2 } - 2 t \right) \mathbf { j }$$ The velocity of \(P\) at time \(t\) seconds is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , \mathbf { v } = \mathbf { i } - 4 \mathbf { j }\).

1. Find \(\mathbf { v }\) at time \(t\) seconds. When \(t = 3\), the particle \(P\) receives an impulse ( \(- 5 \mathbf { i } + 12 \mathbf { j }\) ) N s.
2. Find the speed of \(P\) immediately after it receives the impulse.

1. A particle of mass 0.25 kg is moving with velocity \(( 3 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives the impulse \(( 5 \mathbf { i } - 3 \mathbf { j } )\) N s.

Find the speed of the particle immediately after the impulse.

1. A particle \(P\) of mass 2 kg is moving with velocity \(( \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse of \(( 3 \mathbf { i } + 6 \mathbf { j } ) \mathrm { N } \mathrm { s }\).

Find the speed of \(P\) immediately after the impulse is applied.
(5)

6 Two particles \(A\) and \(B\), of masses \(m \mathrm {~kg}\) and 1 kg respectively, are connected by a light inextensible string of length \(d \mathrm {~m}\) and placed at rest on a smooth horizontal plane a distance of \(\frac { 1 } { 2 } d \mathrm {~m}\) apart. \(B\) is then projected horizontally with speed \(v \mathrm {~ms} ^ { - 1 }\) in a direction perpendicular to \(A B\).

1. Show that, at the instant that the string becomes taut, the magnitude of the instantaneous impulse in the string, \(I \mathrm { Ns }\), is given by \(\mathrm { I } = \frac { \sqrt { 3 } \mathrm { mv } } { 2 ( 1 + \mathrm { m } ) }\).
2. Find, in terms of \(m\) and \(v\), the kinetic energy of \(B\) at the instant after the string becomes taut. Give your answer as a single algebraic fraction.
3. In the case where \(m\) is very large, describe, with justification, the approximate motion of \(B\) after the string becomes taut.

1. A ball of mass 0.6 kg bounces against a wall and is given an impulse of \(( 12 \mathbf { i } - 9 \mathbf { j } ) \mathrm { Ns }\) where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors. The velocity of the particle after the impact is \(( 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

Find the velocity of the particle before the impact.
(4 marks)

1 A smooth horizontal surface lies in the \(x - y\) plane. A particle \(P\) of mass 0.5 kg is moving on the surface with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the \(x\)-direction when it is struck by a horizontal blow whose impulse has components - 3.5 N s and 2.4 N s in the \(x\)-direction and \(y\)-direction respectively.

1. Find the components in the \(x\)-direction and the \(y\)-direction of the velocity of \(P\) immediately after the blow. Hence show that the speed of \(P\) immediately after the blow is \(5.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   \(P\) is struck by a second horizontal blow whose impulse is \(\mathbf { I }\).
2. Given that \(P\) 's direction of motion immediately after this blow is parallel to the \(x\)-axis, write down the component of \(\mathbf { I }\) in the \(y\)-direction.

1 A particle \(P\) of mass 0.3 kg is moving on a smooth horizontal surface with speed \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is struck by a horizontal impulse. The magnitude of the impulse is 0.6 Ns .

1. (a) Find the greatest possible speed of \(P\) after the impulse acts.
   (b) Find the least possible speed of \(P\) after the impulse acts.
2. In fact the speed of \(P\) after the impulse acts is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the angle the impulse makes with the original direction of travel of \(P\) and draw a sketch to make this direction clear.

1 A particle \(P\) of mass 0.2 kg is moving on a smooth horizontal surface with speed \(3 \mathrm {~ms} ^ { - 1 }\), when it is struck by an impulse of magnitude \(I\) Ns. The impulse acts horizontally in a direction perpendicular to the original direction of motion of \(P\), and causes the direction of motion of \(P\) to change by an angle \(\alpha\), where \(\tan \alpha = \frac { 5 } { 12 }\).

1. Show that \(I = 0.25\).
2. Find the speed of \(P\) after the impulse acts.

1
\includegraphics[max width=\textwidth, alt={}, center]{c0f31235-80aa-4838-844f-b706de55e7cd-2_285_1096_255_488} A particle \(P\) of mass 0.3 kg is moving with speed \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on a smooth horizontal surface when it is struck by a horizontal impulse. After the impulse acts \(P\) has speed \(0.6 \mathrm {~ms} ^ { - 1 }\) and is moving in a direction making an angle \(30 ^ { \circ }\) with its original direction of motion (see diagram).

1. Find the magnitude of the impulse and the angle its line of action makes with the original direction of motion of \(P\). Subsequently a second impulse acts on \(P\). After this second impulse acts, \(P\) again moves from left to right with speed \(0.4 \mathrm {~ms} ^ { - 1 }\) in a direction parallel to its original direction of motion.
2. State the magnitude of the second impulse, and show the direction of the second impulse on a diagram.