Vertical motion: given height at two times

A question is this type if and only if a particle passes through the same height at two different times and you must find the speed of projection or the height.

4 questions · Moderate -0.4

3.02h Motion under gravity: vector form
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CAIE M1 2023 November Q1
3 marks Moderate -0.3
1 A particle is projected vertically upwards from horizontal ground with a speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle has height \(s \mathrm {~m}\) above the ground at times 3 seconds and 4 seconds after projection. Find the value of \(u\) and the value of \(s\).
Edexcel M1 2012 June Q5
12 marks Standard +0.3
  1. A particle \(P\) is projected vertically upwards from a point \(A\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The point \(A\) is 17.5 m above horizontal ground. The particle \(P\) moves freely under gravity until it reaches the ground with speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Show that \(u = 21\)
    At time \(t\) seconds after projection, \(P\) is 19 m above \(A\).
  2. Find the possible values of \(t\). The ground is soft and, after \(P\) reaches the ground, \(P\) sinks vertically downwards into the ground before coming to rest. The mass of \(P\) is 4 kg and the ground is assumed to exert a constant resistive force of magnitude 5000 N on \(P\).
  3. Find the vertical distance that \(P\) sinks into the ground before coming to rest.
OCR M1 2007 June Q5
11 marks Moderate -0.3
5 A particle \(P\) is projected vertically upwards, from horizontal ground, with speed \(8.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that the greatest height above the ground reached by P is 3.6 m . A particle Q is projected vertically upwards, from a point 2 m above the ground, with speed \(\mathrm { um } \mathrm { s } ^ { - 1 }\). The greatest height abovetheground reached by Q is also 3.6 m .
  2. Find the value of \(u\). It is given that P and Q are projected simultaneously.
  3. Show that, at the instant when P and Q are at the same height, the particles have the same speed and are moving in opposite directions.
OCR MEI Paper 1 2023 June Q1
2 marks Easy -1.2
1 A ball is thrown vertically upwards with a speed of \(8 \mathrm {~ms} ^ { - 1 }\).
Find the times at which the ball is 3 m above the point of projection.