Newton’s laws of motion

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.

Two particles A and B of masses 0.6 kg and 0.8 kg respectively, are attached to the ends of a light inextensible string. Particle A is held at rest at a fixed point and B hangs vertically below A.

Particle A is now released. As the particles fall the air resistance acting on A has a constant magnitude of R N and the air resistance acting on B has a constant magnitude of 0.3 N. The string remains taut throughout the subsequent motion (see diagram).

The downward acceleration of each of the particles is 9.2 m s^-2 and the tension in the string is T N.

By applying Newton’s second law to B, find the value of T.

Hence, or otherwise, find the value of R.

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.

A rectangular block B is at rest on a horizontal surface. A particle P of mass 2.5 kg is placed on the upper surface of B. The particle P is attached to one end of a light inextensible string which passes over a smooth fixed pulley. A particle Q of mass 3 kg is attached to the other end of the string and hangs freely below the pulley. The part of the string between P and the pulley is horizontal (see diagram).

The particles are released from rest with the string taut. It is given that B remains in equilibrium while P moves on the upper surface of B. The tension in the string while P moves on B is 16.8 N.

Find the acceleration of Q while P and B are in contact.

Determine the coefficient of friction between P and B.

In this question the unit vectors i and j are in the directions east and north respectively.

A particle P is moving on a smooth horizontal surface under the action of a single force F N. At time t seconds, where t ⩾ 0, the velocity v ms^-1 of P, relative to a fixed origin O, is given by

v = (1 - 2t)i + (2t² + t - 13)j.

Show that P is never stationary.

Find, in terms of i and j, the acceleration of P at time t.

The mass of P is 0.5 kg.

Determine the magnitude of F when P is moving in the direction of the vector - 2 i + j. Give

your answer correct to 3 significant figures.

When t = 1, P is at the point with position vector .

Determine the bearing of P from O at time t = 1.5.

In this question the unit vectors i and j are in the directions east and north respectively.

At time t seconds, where t ⩾ 0, a particle P of mass 2 kg is moving on a smooth horizontal surface under the action of a constant horizontal force (- 8 i - 54 j)N and a variable horizontal force (4ti + 6(2t - 1)²j)N.

Determine the value of t when the forces acting on P are in equilibrium.

It is given that P is at rest when t = 0.

Determine the speed of P at the instant when P is moving due north.

Determine the distance between the positions of P when t = 0 and t = 3.

In this question the unit vectors i and j are in the directions east and north respectively.

A particle R of mass 2 kg is moving on a smooth horizontal surface under the action of a single horizontal force F N. At time t seconds, the velocity v ms^-1 of R, relative to a fixed origin O, is given by v = (pt² - 3t)i + (8t + q)j, where p and q are constants and p < 0.

Given that when t = 0.5 the magnitude of F is 20, find the value of p.

When t = 0, R is at the point with position vector (2 i - 3 j) m.

Find, in terms of q, an expression for the displacement vector of R at time t.

When t = 1, R is at a point on the line L, where L passes through O and the point with position vector 2 i - 8 j.

Find the value of q.

Find the acceleration vector of the particle.

Find the position vector of the particle after 10 seconds.