Projectiles

In this question you should take the acceleration due to gravity to be 10 m s^-2.

The unit vectors i and j are in a vertical plane with i horizontal and j vertically upwards.

A small ball P is projected from a point O on horizontal ground into the air. When P is in the air, it is to be modelled as a particle moving under the influence of gravity only.

At time t = 2 seconds, P has velocity (2i - 8j) m s^-1. The position vector of a point on the trajectory of P is (x i + y j) m relative to O.

Show that y = 6x - 1.25x².

Determine the maximum height of P above the ground during its motion.

Determine the following as P passes through the point on its trajectory where x = 2.5.

• the speed of P

• the direction of motion of P

In reality P does not move only under the influence of gravity but is also subject to air resistance.

Explain how this would affect your answer to part (b).

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 small ball B moves in the plane of a fixed horizontal axis Ox, which lies on horizontal ground, and a fixed vertically upwards axis Oy. B is projected from O with a velocity whose components along Ox and Oy are U ms^-1 and V ms^-1, respectively. The units of x and y are metres.

B is modelled as a particle moving freely under gravity.

Show that the path of B has equation 2U²y = 2UVx - gx².

During its motion, B just clears a vertical wall of height m at a horizontal distance a m from O. B strikes the ground at a horizontal distance 3a m beyond the wall.

Determine the angle of projection of B. Give your answer in degrees correct to 3 significant

figures.

Given that the speed of projection of B is 54.6 ms^-1, determine the value of a.

Hence find the maximum height of B above the ground during its motion.

State one refinement of the model, other than including air resistance, that would make it

more realistic.

A golfer hits a ball from a point A with a speed of 25 m s^-1 at an angle of 15° above the horizontal. While the ball is in the air, it is modelled as a particle moving under the influence of gravity. Take the acceleration due to gravity to be 10 m s^-2.

The ball first lands at a point B which is 4 m below the level of A (see diagram).

Determine the time taken for the ball to travel from A to B.

Determine the horizontal distance of B from A.

Determine the direction of motion of the ball 1.5 seconds after the golfer hits the ball.

The horizontal distance from A to B is found to be greater than the answer to part (b).

State one factor that could account for this difference.

A particle P moves freely under gravity in the plane of a fixed horizontal axis Ox, which lies on horizontal ground, and a fixed vertical axis Oy. P is projected from O with a velocity whose components along Ox and Oy are U and V, respectively. P returns to the ground at a point C.

Determine, in terms of U, V and g, the distance OC.

P passes through two points A and B, each at a height h above the ground and a distance d apart, as shown in the diagram.

Write down the horizontal and vertical components of the velocity of P at A.

Hence determine an expression for d in terms of U, V, g and h.

Given that the direction of motion of P as it passes through A is inclined to the horizontal at an

angle θ, where tan θ = , determine an expression for V in terms of g, d and h.

A particle P projected from a point O on horizontal ground hits the ground after 2.4 seconds.

The horizontal component of the initial velocity of P is ms^-1.

Find, in terms of d, the horizontal distance of P from O when it hits the ground.

Find the vertical component of the initial velocity of P.

P just clears a vertical wall which is situated at a horizontal distance d m from O.

Find the height of the wall.

The speed of P as it passes over the wall is 16ms^-1.

Find the value of d correct to 3 significant figures.