Impulse & momentum

A particle P of mass 4 kg is moving in a straight line with a speed of 7 m s^-1 on a smooth horizontal surface.

A constant force of magnitude 12 N acts on P for 5 seconds in the same direction as P ’s motion.

Find the magnitude of the impulse exerted by the force on P. You must include the units.

Hence determine the speed of P at the instant when the force stops acting.

By considering the change in the kinetic energy of P, determine the work done by the force

over the 5 seconds that it acts.

Hence find the distance travelled by P while the force is acting on it.

One end of a light inextensible string of length r m is attached to a fixed point O. The other end is attached to a particle P of mass m kg.

Particle P hangs in equilibrium vertically below O with the string taut. An initial horizontal impulse is applied to P so that it instantaneously starts moving horizontally with a speed of u m s^-1.

Particle P subsequently moves in complete circles in a vertical plane containing O.

When the line OP makes an angle of θ with the downward vertical through O, the speed of P is v m s^-1 and the tension in the string is TN.

Show that T = u² + α gr + β gr cos θ) where α and β are constants to be determined.

Determine, in terms of m, g and r, the minimum initial impulse required to ensure that P

describes complete vertical circles around O with the string remaining taut.

A particle P of mass 12.5 kg is moving on a smooth horizontal plane when it collides obliquely with a fixed vertical wall.

At the instant before the collision, the velocity of P is -5i + 12j ms^-1.

At the instant after the collision, the velocity of P is i + 4j ms^-1.

Find the magnitude of the momentum of P before the collision.

Find, in vector form, the impulse that the wall exerts on P.

State, in vector form, the impulse that P exerts on the wall.

Find in either order.

• The magnitude of the impulse that the wall exerts on P.

• The angle between i and the impulse that the wall exerts on P.

A particle P of mass 6 kg moves in a straight line under the action of a single force of magnitude F N which acts in the direction of motion of P. At time t seconds, where t ⩾ 0, F is given by F = When t = 0, the speed of P is 1.9 m s^-1.

Find the impulse of the force over the period 0 ⩽ t ⩽ 2.

Find the speed of P at the instant when t = 2.

Find the work done by the force on P over the period 0 ⩽ t ⩽ 2.

A particle P of mass 2.5 kg is free to move along the x ‑axis. When its displacement from the origin is x m its velocity is v m s^-1.

At time t = 0 seconds, P is at the point where x = 1 and is travelling in the negative x ‑direction with speed 5 m s^-1.

At this time an impulse of I Ns is applied to P in the positive x ‑direction so that P moves in the positive x‑direction with speed 18 m s^-1.

Find the value of I.

Subsequently, whenever P is in motion, two forces act on it. The first force acts in the positive x‑direction and has magnitude N. The second force acts in the negative x‑direction and has magnitude 60 v N.

Show that the motion of P can be modelled by the differential equation = + b where a

and b are constants whose values should be determined.

By solving the differential equation derived in part (b) find an expression for v in terms of x.

Describe in detail the motion of P when t ⩾ 0.