Vector impulse: find deflection angle or impulse magnitude from angle

1. A particle \(P\), of mass 0.5 kg , is moving with velocity ( \(4 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse I of magnitude 2.5 Ns.

As a result of the impulse, the direction of motion of \(P\) is deflected through an angle of \(45 ^ { \circ }\) Given that \(\mathbf { I } = ( \lambda \mathbf { i } + \mu \mathbf { j } )\) Ns, find all the possible pairs of values of \(\lambda\) and \(\mu\).

1. A particle \(P\) has mass 0.5 kg . It is moving in the \(x y\) plane with velocity \(8 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives an impulse \(\lambda ( - \mathbf { i } + \mathbf { j } )\) Ns, where \(\lambda\) is a positive constant.

The angle between the direction of motion of \(P\) immediately before receiving the impulse and the direction of motion of \(P\) immediately after receiving the impulse is \(\theta ^ { \circ }\) Immediately after receiving the impulse, \(P\) is moving with speed \(4 \sqrt { 10 } \mathrm {~ms} ^ { - 1 }\) Find (i) the value of \(\lambda\) (ii) the value of \(\theta\)

\includegraphics{figure_1} The points \(A\), \(B\) and \(C\) lie in a horizontal plane. A batsman strikes a ball of mass \(0.25\) kg. Immediately before being struck, the ball is moving along the horizontal line \(AB\) with speed \(30 \text{ ms}^{-1}\). Immediately after being struck, the ball moves along the horizontal line \(BC\) with speed \(40 \text{ ms}^{-1}\). The line \(BC\) makes an angle of \(60°\) with the original direction of motion \(AB\), as shown in Figure 1. Find, to 3 significant figures,

1. the magnitude of the impulse given to the ball,
2. the size of the angle that the direction of this impulse makes with the original direction of motion \(AB\).

[8]

A tennis ball of mass \(0.2\) kg is moving with velocity \((-10\mathbf{i})\) m s\(^{-1}\) when it is struck by a tennis racket. Immediately after being struck, the ball has velocity \((15\mathbf{i}+ 15\mathbf{j})\) m s\(^{-1}\). Find

1. the magnitude of the impulse exerted by the racket on the ball, [4]
2. the angle, to the nearest degree, between the vector \(\mathbf{i}\) and the impulse exerted by the racket, [2]
3. the kinetic energy gained by the ball as a result of being struck. [2]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in a horizontal plane.] A ball of mass 0.5 kg is moving with velocity \((10\mathbf{i} + 24\mathbf{j})\) m s\(^{-1}\) when it is struck by a bat. Immediately after the impact the ball is moving with velocity \(20\mathbf{i}\) m s\(^{-1}\). Find

1. the magnitude of the impulse of the bat on the ball, (4)
2. the size of the angle between the vector \(\mathbf{i}\) and the impulse exerted by the bat on the ball, (2)
3. the kinetic energy lost by the ball in the impact. (3)

A small ball of mass \(0.8\) kg is moving with speed \(10.5\) m s\(^{-1}\) when it receives an impulse of magnitude \(4\) N s. The speed of the ball immediately afterwards is \(8.5\) m s\(^{-1}\). The angle between the directions of motion before and after the impulse acts is \(\alpha\). Using an impulse-momentum triangle, or otherwise, find \(\alpha\). [6]

\includegraphics{figure_1} A particle \(P\) of mass \(0.3\) kg is moving in a straight line with speed \(4\) m s\(^{-1}\) when it is deflected through an angle \(\theta\) by an impulse of magnitude \(I\) N s. The impulse acts at right angles to the initial direction of motion of \(P\) (see diagram). The speed of \(P\) immediately after the impulse acts is \(5\) m s\(^{-1}\). Show that \(\cos \theta = 0.8\) and find the value of \(I\). [4]

A particle \(P\) of mass \(0.2\) kg is moving on a smooth horizontal surface with speed \(3\text{ ms}^{-1}\), 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 \(I = 0.25\). [4]
2. Find the speed of \(P\) after the impulse acts. [2]

\includegraphics{figure_1} A particle \(P\) of mass \(0.3\) kg is moving with speed \(0.4\) m 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 \(0.6\) m s\(^{-1}\) and is moving in a direction making an angle \(30°\) 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 \(P\). [4]

Subsequently a second impulse acts on \(P\). After this second impulse acts, \(P\) again moves from left to right with speed \(0.4\) m s\(^{-1}\) in a direction parallel to its original direction of motion.

1. State the magnitude of the second impulse, and show the direction of the second impulse on a diagram. [2]

A ball has mass 0.4 kg and is hit by a wooden bat. The speed of the ball just before it is hit by the bat is \(6 \text{ m s}^{-1}\) The velocity of the ball immediately after being hit by the bat is perpendicular to its initial velocity. The speed of the ball just after it is hit by the bat is \(8 \text{ m s}^{-1}\) Show that the impulse on the ball has magnitude 4 N s [3 marks]