A fixed smooth plane is inclined to the horizontal at an angle of 45°. A particle P is moving horizontally and strikes the plane. Immediately before the impact, P is moving in a vertical plane containing a line of greatest slope of the inclined plane. Immediately after the impact, P is moving in a direction which makes an angle of 30° with the inclined plane, as shown in Figure 1. Find the fraction of the kinetic energy of P which is lost in the impact.
(6)
k is a constant. At time t the speed of P is v. The particle P moves against air resistance whose magnitude is modelled as being mkv2 when the speed of P is v. Find, in terms of k, the distance travelled by P until its speed first becomes half of its initial speed.
(9)
At noon a motorboat P is 2 km north-west of another motorboat Q. The motorboat P is moving due south at 20 m s–1. The motorboat Q is pursuing motorboat P at a speed of 12 m s–1 and sets a course in order to get as close to motorboat P as possible.
Find the course set by Q, giving your answer as a bearing to the nearest degree.
(4)
Find the shortest distance between P and Q.
(3)
Find the distance travelled by Q from its position at noon to the point of closest approach.
(5)
A light inextensible string of length 2 a has one end attached to a fixed point A. The other end of the string is attached to a particle P of mass m. A second light inextensible string of length L, where L > 
, has one of its ends attached to P and passes over a small smooth peg fixed at a point B. The line AB is horizontal and AB = 2a. The other end of the second string is attached to a particle of mass 
m, which hangs vertically below B, as shown in Figure 2.
Show that the potential energy of the system, when the angle PAB = 2 θ, is 
mga(7sin
10sin 2
) + constant.
(4)
Show that there is only one value of cos θ for which the system is in equilibrium and find this value.
(8)
Determine the stability of the position of equilibrium.
(4)
Two small smooth spheres A and B, of mass 2 kg and 1 kg respectively, are moving on a smooth horizontal plane when they collide. Immediately before the collision the velocity of A is (i + 2j) m s–1 and the velocity of B is –2i m s–1. Immediately after the collision the velocity of A is j m s–1.
Show that the velocity of B immediately after the collision is 2j m s–1.
(3)
Find the impulse of B on A in the collision, giving your answer as a vector, and hence show that the line of centres is parallel to i + j.
(4)
Find the coefficient of restitution between A and B.
(6)
A light elastic spring AB has natural length 2a and modulus of elasticity 2mn2a, where n is a constant. A particle P of mass m is attached to the end A of the spring. At time t = 0, the spring, with P attached, lies at rest and unstretched on a smooth horizontal plane. The other end B of the spring is then pulled along the plane in the direction AB with constant acceleration f. At time t the extension of the spring is x.
Hence find
the maximum extension of the spring,
(3)
the speed of P when the spring first reaches its maximum extension.
(2)
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