13 A ball falls freely towards the Earth.
The ball passes through two different fixed points \(M\) and \(N\) before reaching the Earth's surface.
At \(M\) the ball has velocity \(u \mathrm {~ms} ^ { - 1 }\)
At \(N\) the ball has velocity \(3 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
It can be assumed that:
- the motion is due to gravitational force only
- the acceleration due to gravity remains constant throughout.
13
- Show that the time taken for the ball to travel from \(M\) to \(N\) is \(\frac { 2 u } { g }\) seconds.
[0pt]
[2 marks]
13 - Point \(M\) is \(h\) metres above the Earth.
Show that \(h > \frac { 4 u ^ { 2 } } { g }\)
Fully justify your answer.
The car is moving in a straight line.
The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the car at time \(t\) seconds is given by
$$a = 3 k t ^ { 2 } - 2 k t + 1$$
where \(k\) is a constant.
When \(t = 3\) the car has a velocity of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
Show that \(k = \frac { 1 } { 3 }\)