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Equations of Motion

Q1.

(a) Indicate with a tick in the correct box which of the following quantities are vectors and which are scalars.

  velocity speed distance displacement
scalar
vector

One mark for each correct row of ticks.

[2 marks]

(b) A tennis ball is thrown vertically downwards and bounces on the ground. The ball leaves the hand with an initial speed of 1.5 m s–1 at a height of 0.65 m above the ground. The ball rebounds and is caught when travelling upwards with a speed of 1.0 m s–1. Assume that air resistance is negligible.

(i) Show that the speed of the ball is about 4 m s–1 just before it strikes the ground.

u = 1.5 m s–1

v = ?

a = 9.81 m s–2

s = 0.65 m

v2 = u2 + 2as

v2 = 1.52 + 2 x 9.81 x 0.65 = 15.0

v = 3.87 m/s

v = 3.9 m/s (final answer should be given to 2sf as the values in the question are give to 2sf) which is about 4 m/s.

[3 marks]

(ii) The ball is released at time t = 0. It hits the ground at time tA and is caught at time tB. On the axes provided, sketch a velocity–time graph for the vertical motion of the tennis ball from when it leaves the hand to when it returns.

The initial velocity X and final velocity Y are marked on the graph for you.

[3 marks]

(c) In a game of tennis, a ball is hit horizontally at a height of 1.2 m and travels a horizontal distance of 5.0 m before reaching the ground.

The ball is at rest when hit.

Calculate the initial horizontal velocity given to the ball when it was hit.

Horizontally the ball will travel at a constant velocity, for a distance of 5.0 m until it hits the ground.

We need to find that velocity by using v = s/t, but we don't know the time.

For the time it is travelling horizontally at a steady velocity it is accelerating in the vertical direction.

Consider the vertical motion:

u = 0 m/s

s = 1.2 m

a = 9.81 m s–2

t = ?

s = ut + 1/2 at2

1.2 = 0 + 0.5 x 9.81 x t2

t2 = 1.2 x 2/9.81 = 0.245

t = 0.495 seconds

(I have written 'seconds' in full here because 's' stands for displacement and I don't want to confuse myself or the examiner!)

Consider the horizontal motion:

s = 5.0 m

t = 0.495 seconds

v = s/t

v = 5.0/0.495 = 10 m/s

[3 marks]

(Total 11 marks)