June 2023 Paper 2 Q13
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 \(u\) m s−1
At \(N\) the ball has velocity \(3u\) m s−1
It can be assumed that:
- the motion is due to gravitational force only
- the acceleration due to gravity remains constant throughout.
(a) Show that the time taken for the ball to travel from \(M\) to \(N\) is \(\dfrac{2u}{g}\) seconds. [2 marks]
(b) Point \(M\) is \(h\) metres above the Earth.
Show that \(h \gt \dfrac{4u^2}{g}\)
Fully justify your answer. [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| Selects an appropriate equation of constant acceleration to find \(t\) and uses \(u = u\) and \(v = 3u\) | M1 | 1.1a |
| Completes reasoned argument show the given result. Must have clearly stated \(u = u \quad v = 3u \quad a = g\) and must see either \(3u = u + gt\) or \(\dfrac{3u - u}{g} = t\) AG | R1 | 2.1 |
| (2) |
Typical solution
\[v = u + at\]\[u = u \qquad v = 3u \qquad a = g\]\[3u = u + gt\]\[\frac{3u - u}{g} = t\]\[t = \frac{2u}{g}\]| Scheme | Marks | AO |
|---|---|---|
| Selects a correct equation of constant acceleration to find \(s\) and substitutes correctly. Condone \(a = -g\) | M1 | 3.3 |
| Completes reasoned argument with at least one more intermediate step to obtain \(\dfrac{4u^2}{g}\) | A1 | 1.1b |
| Explains that have found the distance \(MN\) and \(N\) is not on the surface to justify \(h \gt \dfrac{4u^2}{g}\) AG | R1 | 2.4 |
| (3) | ||
| (5 marks) |
Typical solution
\[v^2 = u^2 + 2as\]\[u = u \qquad v = 3u \qquad a = g\]\[(3u)^2 = u^2 + 2gs\]\[9u^2 = u^2 + 2gs\]\[8u^2 = 2gs\]\[MN = s = \frac{4u^2}{g}\]Since \(N\) is above the ground then
\[h \gt \frac{4u^2}{g}\]