Vertical projection: time at height

A question is this type if and only if you must find the time duration for which a particle is above (or below) a specified height during vertical motion.

7 questions · Moderate -0.4

3.02d Constant acceleration: SUVAT formulae3.02h Motion under gravity: vector form
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CAIE M1 2021 June Q4
6 marks Moderate -0.3
4 A particle is projected vertically upwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on horizontal ground. After 2 seconds, the height of the particle above the ground is 24 m .
  1. Show that \(u = 22\).
  2. The height of the particle above the ground is more than \(h \mathrm {~m}\) for a period of 3.6 s . Find \(h\).
CAIE M1 2013 June Q3
7 marks Standard +0.3
3 The top of a cliff is 40 metres above the level of the sea. A man in a boat, close to the bottom of the cliff, is in difficulty and fires a distress signal vertically upwards from sea level. Find
  1. the speed of projection of the signal given that it reaches a height of 5 m above the top of the cliff,
  2. the length of time for which the signal is above the level of the top of the cliff. The man fires another distress signal vertically upwards from sea level. This signal is above the level of the top of the cliff for \(\sqrt { } ( 17 ) \mathrm { s }\).
  3. Find the speed of projection of the second signal.
CAIE M1 2019 June Q2
7 marks Moderate -0.5
2 A particle \(P\) is projected vertically upwards with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 3 m above horizontal ground.
  1. Find the time taken for \(P\) to reach its greatest height.
  2. Find the length of time for which \(P\) is higher than 23 m above the ground.
  3. \(P\) is higher than \(h \mathrm {~m}\) above the ground for 1 second. Find \(h\).
Edexcel M1 2012 January Q5
11 marks Moderate -0.8
  1. A stone is projected vertically upwards from a point \(A\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After projection the stone moves freely under gravity until it returns to \(A\). The time between the instant that the stone is projected and the instant that it returns to \(A\) is \(3 \frac { 4 } { 7 }\) seconds.
Modelling the stone as a particle,
  1. show that \(u = 17 \frac { 1 } { 2 }\),
  2. find the greatest height above \(A\) reached by the stone,
  3. find the length of time for which the stone is at least \(6 \frac { 3 } { 5 } \mathrm {~m}\) above \(A\).
Edexcel M1 2015 June Q2
7 marks Moderate -0.8
2. A small stone is projected vertically upwards from a point \(O\) with a speed of \(19.6 \mathrm {~ms} ^ { - 1 }\). Modelling the stone as a particle moving freely under gravity,
  1. find the greatest height above \(O\) reached by the stone,
  2. find the length of time for which the stone is more than 14.7 m above \(O\).
Edexcel AS Paper 2 2023 June Q2
7 marks Moderate -0.3
  1. A small stone is projected vertically upwards with speed \(39.2 \mathrm {~ms} ^ { - 1 }\) from a point \(O\).
The stone is modelled as a particle moving freely under gravity from when it is projected until it hits the ground 10s later. Using the model, find
  1. the height of \(O\) above the ground,
  2. the total length of time for which the speed of the stone is less than or equal to \(24.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  3. State one refinement that could be made to the model that would make your answer to part (a) more accurate.
Edexcel M1 Q6
10 marks Moderate -0.3
6. A boy kicks a football vertically upwards from a height of 0.6 m above the ground with a speed of \(10.5 \mathrm {~ms} ^ { - 1 }\). The ball is modelled as a particle and air resistance is ignored.
  1. Find the greatest height above the ground reached by the ball.
  2. Calculate the length of time for which the ball is more than 2 m above the ground.