A particle moving in a straight line with constant acceleration passes three points p, q, r and has speeds u and 7u at p and r respectively.

1. Find its speed at the midpoint of [pr] in terms of u.
2. Show that the time from p to q is twice that from q to r.

Between p and q v² = u² + 2ax

Between q and r (7u)² = v² + 2ax

Then getting rid of 2ax gives:

v² = u² + 2ax

49u² = v² + 2ax

• 49u² – v² = v² – u²
• v = 5u

(ii) Applying the formula v = u + at to each half:

Between p and q: 5u = u + at₁

Between q and r: 7u = 5u + at₂

A juggler throws up six balls, one after the other at equal intervals of time t, each to a height of 3m. The first ball returns to his hand t seconds after the sixth was thrown up and is immediately thrown to the same height, and so on continually. (assume that each ball moves vertically).

Find

1. the initial velocity of each ball
2. the time t
3. the heights of the other balls when any one reaches the juggler’s hand.