Constant acceleration formulae

A particle P of mass 3 kg is moving on a smooth horizontal surface under the action of two constant horizontal forces (5i + 3j)N and (ai + 3bj)N. The acceleration of P is (2i - 3j) m s^-2.

Find the value of a and the value of b.

At time t = 0 seconds the velocity of P is u m s^-1 and at time t = 5 seconds the velocity of P is (7i - 6j) m s^-1.

Find, in terms of i and j, an expression for u.

A particle P is moving with constant acceleration (-5i + 2j)m s^-2. At time t = 0 seconds, P is at the origin and has velocity (i + 3j)m s^-1.

Find, in terms of i and j, the displacement of P at time t = 2 seconds.

Determine the speed of P at time t = 2 seconds.

The points A and B are the lower and upper ends, respectively, of a line of greatest slope on a plane inclined at an angle θ to the horizontal, where sin θ = 0.6 and AB = 1.375 m (see diagram).

A particle P is projected up the plane with speed 6 ms^-1 from A towards B.

The plane at A is fixed to the ground which is horizontal.

The surface of the plane is rough and the coefficient of friction between P and the plane is 0.25.

Show that the speed of P at B is 3.8 ms^-1.

The particle leaves the slope at B and moves freely under gravity.

The particle first lands at a point C on the horizontal ground. The time taken for P to travel from A to C is T seconds.

Determine the value of T.

There are three checkpoints, A, B and C, in that order, on a straight horizontal road. A car travels along the road, in the direction from A to C, with constant acceleration. The car takes 20 s to travel from B to C. The speed of the car at B is 14 m s^-1 and the speed of the car at C is 18 m s^-1.

Find the acceleration of the car.

It is given that the distance between A and B is 330 m.

Determine the speed of the car at A.

A particle P moves with constant acceleration (-4i + 2j)ms^-2. At time t = 0 seconds, P is moving with velocity (7i + 6j)ms^-1.

Determine the speed of P when t = 3.

Determine the change in displacement of P between t = 0 and t = 3.

One end of a light inextensible string is attached to a particle A of mass 2 kg. The other end of the string is attached to a second particle B of mass 2.5 kg. Particle A is in contact with a rough plane inclined at θ to the horizontal, where cos θ = . The string is taut and passes over a small smooth pulley P at the top of the plane. The part of the string from A to P is parallel to a line of greatest slope of the plane. Particle B hangs freely below P at a distance 1.5 m above horizontal ground, as shown in the diagram.

The coefficient of friction between A and the plane is μ. The system is released from rest and in the subsequent motion B hits the ground before A reaches P. The speed of B at the instant that it hits the ground is 1.2 ms^-1.

For the motion before B hits the ground, show that the acceleration of B is 0.48 ms^-2.

For the motion before B hits the ground, show that the tension in the string is 23.3 N.

Determine the value of μ.

After B hits the ground, A continues to travel up the plane before coming to instantaneous rest before it reaches P.

Determine the distance that A travels from the instant that B hits the ground until A comes to

instantaneous rest.

The diagram shows a small block B, of mass 0.2 kg, and a particle P, of mass 0.5 kg, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane.

The block can move on the horizontal surface, which is rough. The particle can move on the inclined plane, which is smooth and which makes an angle of θ with the horizontal where tan θ =

The system is released from rest. In the first 0.4 seconds of the motion P moves 0.3 m down the plane and B does not reach the pulley.

Find the tension in the string during the first 0.4 seconds of the motion.

Calculate the coefficient of friction between B and the horizontal surface.

Find the acceleration vector of the particle.

Find the position vector of the particle after 10 seconds.

The velocity v m s^-1 of a car at time t s, during the first 20 s of its journey, is given by v = kt + 0.03t², where k is a constant. When t = 20 the acceleration of the car is 1.3 m s^-2. For t > 20 the car continues its journey with constant acceleration 1.3 m s^-2 until its speed reaches 25 m s^-1.

Find the value of k.

Find the total distance the car has travelled when its speed reaches 25 m s^-1.