Vertical motion under gravity

A question is this type if and only if a particle is projected vertically upwards or falls freely under gravity (possibly through a liquid or from a height), and the task involves finding speed, time, or distance using constant acceleration due to gravity.

15 questions · Moderate -0.8

3.02d Constant acceleration: SUVAT formulae3.02h Motion under gravity: vector form
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OCR M1 2011 January Q3
10 marks Moderate -0.8
3 A particle is projected vertically upwards with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 2.5 m above the ground.
  1. Calculate the speed of the particle when it strikes the ground.
  2. Calculate the time after projection when the particle reaches the ground.
  3. Sketch on separate diagrams
    1. the \(( t , v )\) graph,
    2. the \(( t , x )\) graph,
      representing the motion of the particle.
OCR M1 2012 June Q2
8 marks Moderate -0.8
2 A particle is projected vertically upwards with speed \(7 \mathrm {~ms} ^ { - 1 }\) from a point on the ground.
  1. Find the speed of the particle and its distance above the ground 0.4 s after projection.
  2. Find the total distance travelled by the particle in the first 0.9 s after projection.
OCR M1 2013 June Q2
8 marks Moderate -0.8
2 A particle \(P\) is projected vertically upwards and reaches its greatest height 0.5 s after the instant of projection. Calculate
  1. the speed of projection of \(P\),
  2. the greatest height of \(P\) above the point of projection. It is given that the point of projection is 0.539 m above the ground.
  3. Find the speed of \(P\) immediately before it strikes the ground.
OCR M1 2015 June Q1
7 marks Moderate -0.8
1 A particle \(P\) is projected vertically downwards with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 30 m above the ground.
  1. Calculate the speed of \(P\) when it reaches the ground.
  2. Find the distance travelled by \(P\) in the first 0.4 s of its motion.
  3. Calculate the time taken for \(P\) to travel the final 15 m of its descent.
OCR MEI M1 2010 June Q1
3 marks Easy -1.2
1 An egg falls from rest a distance of 75 cm to the floor.
Neglecting air resistance, at what speed does it hit the floor?
OCR MEI M1 2011 June Q1
3 marks Easy -1.2
1 A pellet is fired vertically upwards at a speed of \(11 \mathrm {~ms} ^ { - 1 }\). Assuming that air resistance may be neglected, calculate the speed at which the pellet hits a ceiling 2.4 m above its point of projection.
OCR M1 2014 June Q1
7 marks Moderate -0.3
1 A particle \(P\) is projected vertically downwards with initial speed \(3.5 \mathrm {~ms} ^ { - 1 }\) from a point \(A\) which is 5 m above horizontal ground.
  1. Find the speed of \(P\) immediately before it strikes the ground. After striking the ground, \(P\) rebounds and moves vertically upwards and 0.87 s after leaving the ground \(P\) passes through \(A\).
  2. Calculate the speed of \(P\) immediately after it leaves the ground. It is given that the mass of \(P\) is 0.2 kg .
  3. Calculate the change in the momentum of \(P\) as a result of its collision with the ground.
Edexcel AS Paper 2 2020 June Q1
12 marks Moderate -0.8
  1. At time \(t = 0\), a small ball is projected vertically upwards with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) that is 16.8 m above horizontal ground.
The speed of the ball at the instant immediately before it hits the ground for the first time is \(19 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The ball hits the ground for the first time at time \(t = T\) seconds.
The motion of the ball, from the instant it is projected until the instant just before it hits the ground for the first time, is modelled as that of a particle moving freely under gravity. The acceleration due to gravity is modelled as having magnitude \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Using the model,
  1. show that \(U = 5\)
  2. find the value of \(T\),
  3. find the time from the instant the ball is projected until the instant when the ball is 1.2 m below \(A\).
  4. Sketch a velocity-time graph for the motion of the ball for \(0 \leqslant t \leqslant T\), stating the coordinates of the start point and the end point of your graph. In a refinement of the model of the motion of the ball, the effect of air resistance on the ball is included and this refined model is now used to find the value of \(U\).
  5. State, with a reason, how this new value of \(U\) would compare with the value found in part (a), using the initial unrefined model.
  6. Suggest one further refinement that could be made to the model, apart from including air resistance, that would make the model more realistic.
Edexcel AS Paper 2 2022 June Q1
7 marks Moderate -0.8
  1. The point \(A\) is 1.8 m vertically above horizontal ground.
At time \(t = 0\), a small stone is projected vertically upwards with speed \(U \mathrm {~ms} ^ { - 1 }\) from the point \(A\). At time \(t = T\) seconds, the stone hits the ground.
The speed of the stone as it hits the ground is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) In an initial model of the motion of the stone as it moves from \(A\) to where it hits the ground
  • the stone is modelled as a particle moving freely under gravity
  • the acceleration due to gravity is modelled as having magnitude \(\mathbf { 1 0 } \mathbf { m ~ s } ^ { \mathbf { - 2 } }\)
Using the model,
  1. find the value of \(U\),
  2. find the value of \(T\).
  3. Suggest one refinement, apart from including air resistance, that would make the model more realistic. In reality the stone will not move freely under gravity and will be subject to air resistance.
  4. Explain how this would affect your answer to part (a).
Edexcel M1 Q1
3 marks Moderate -0.3
  1. A car accelerates from 0 to \(108 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in 7.5 seconds. Find its acceleration in \(\mathrm { ms } ^ { - 2 }\). ( 3 marks)
  2. A book rests on a rough desk-lid which is hinged at one end. When the lid is raised so that it makes an angle of \(15 ^ { \circ }\) with the horizontal, the book is just on the point of sliding down a line of greatest slope. Modelling the book as a particle, find
    1. the coefficient of friction between the book and the desk-lid,
    2. the acceleration with which the book starts to move if it is released from rest when the lid is inclined at \(20 ^ { \circ }\) to the horizontal.
    3. A particle \(P\) is projected vertically upwards from ground level at time \(t = 0\) with speed 20 \(\mathrm { ms } ^ { - 1 }\). Two seconds later another particle \(Q\) is projected vertically upwards with speed 30 \(\mathrm { ms } ^ { - 1 }\) from a point on the same horizontal ground.
    4. Taking the upward direction as positive, write down expressions in terms of \(g\) and \(t\) for the velocities of \(P\) and of \(Q\) at time \(t\) seconds after \(P\) is projected.
    5. Find the value of \(t\) when both particles are moving with the same speed.
    6. A jet of water issues from a cylindrical pipe with a circular cross-section of radius \(2 \cdot 75 \mathrm {~cm}\). The water strikes a vertical wall at a speed of \(9 \mathrm {~ms} ^ { - 1 }\). Taking the density of water to be \(1000 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\), calculate
    7. the momentum destroyed each second by the impact with the wall,
    8. the magnitude of the force exerted by the water on the wall.
    9. State one modelling assumption that you have made.
    10. Two particles \(A\) and \(B\), of mass 1 kg and \(m \mathrm {~kg}\) respectively, where \(m > 1\), are attached to the ends of a light inextensible string which passes over a small fixed smooth pulley. The particles are released from rest and move with the string taut and vertical.
    11. Show that the acceleration of the system is equal to \(\frac { ( m - 1 ) g } { m + 1 }\).
    12. Find the tension in the string, in terms of \(m\) and \(g\), expressing your answer as a single algebraic fraction in its simplest form.
    When the system is released from rest, both particles are \(52 \cdot 5 \mathrm {~cm}\) above ground level and 60 cm below the level of the pulley. \(B\) hits the ground after half a second.
  3. Find the value of \(m\).
  4. Find the speed with which \(B\) hits the ground. \section*{MECHANICS 1 (A) TEST PAPER 9 Page 2}
Edexcel M1 2017 October Q5
11 marks Moderate -0.8
A small ball is projected vertically upwards from a point \(O\) with speed 14.7 m s\(^{-1}\). The point \(O\) is 2.5 m above the ground. The motion of the ball is modelled as that of a particle moving freely under gravity. Find
  1. the maximum height above the ground reached by the ball, [4]
  2. the time taken for the ball to first reach a height of 1 m above the ground, [4]
  3. the speed of the ball at the instant before it strikes the ground for the first time. [3]
Edexcel M1 2022 October Q5
9 marks Moderate -0.3
A small ball is projected vertically upwards with speed \(29.4\text{ ms}^{-1}\) from a point \(A\) which is \(19.6\text{ m}\) above horizontal ground. The ball is modelled as a particle moving freely under gravity until it hits the ground. It is assumed that the ball does not rebound.
  1. Find the distance travelled by the ball while its speed is less than \(14.7\text{ ms}^{-1}\) [3]
  2. Find the time for which the ball is moving with a speed of more than \(29.4\text{ ms}^{-1}\) [3]
  3. Sketch a speed-time graph for the motion of the ball from the instant when it is projected from \(A\) to the instant when it hits the ground. Show clearly where your graph meets the axes. [3]
Edexcel M1 Specimen Q6
10 marks Moderate -0.8
A ball is projected vertically upwards with a speed of 14.7 m s\(^{-1}\) from a point which is 49 m above horizontal ground. Modelling the ball as a particle moving freely under gravity, find
  1. the greatest height, above the ground, reached by the ball, [4]
  2. the speed with which the ball first strikes the ground, [3]
  3. the total time from when the ball is projected to when it first strikes the ground. [3]
Edexcel M1 2003 January Q7
14 marks Moderate -0.3
A ball is projected vertically upwards with a speed \(u\) m s\(^{-1}\) from a point \(A\) which is 1.5 m above the ground. The ball moves freely under gravity until it reaches the ground. The greatest height attained by the ball is 25.6 m above \(A\).
  1. Show that \(u = 22.4\). [3]
The ball reaches the ground 7 seconds after it has been projected from \(A\).
  1. Find, to 2 decimal places, the value of \(T\). [4]
The ground is soft and the ball sinks 2.5 cm into the ground before coming to rest. The mass of the ball is 0.6 kg. The ground is assumed to exert a constant resistive force of magnitude \(F\) newtons.
  1. Find, to 3 significant figures, the value of \(F\). [6]
  2. State one physical factor which could be taken into account to make the model used in this question more realistic. [1]
AQA AS Paper 1 2018 June Q11
1 marks Easy -2.0
In this question use \(g = 9.8\,\mathrm{m}\,\mathrm{s}^{-2}\) A ball, initially at rest, is dropped from a height of \(40\,\mathrm{m}\) above the ground. Calculate the speed of the ball when it reaches the ground. Circle your answer. [1 mark] \(-28\,\mathrm{m}\,\mathrm{s}^{-1}\) \quad \(28\,\mathrm{m}\,\mathrm{s}^{-1}\) \quad \(-780\,\mathrm{m}\,\mathrm{s}^{-1}\) \quad \(780\,\mathrm{m}\,\mathrm{s}^{-1}\)