SUVAT in 2D & Gravity

1 A particle is thrown vertically upwards and returns to its point of projection after 6 seconds. Air resistance is negligible. Calculate the speed of projection of the particle and also the maximum height it reaches.

1 A particle \(P\) is projected vertically upwards with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on the ground.

1. Find the greatest height above the ground reached by \(P\).
2. Find the total time from projection until \(P\) returns to the ground.

A particle \(P\) of mass \(0.4\) kg is projected vertically upwards from horizontal ground with speed \(10\) m s\(^{-1}\).

1. Find the greatest height above the ground reached by \(P\). [2]

When \(P\) reaches the ground again, it bounces vertically upwards. At the first instant that it hits the ground, \(P\) loses \(7.2\) J of energy.

1. Find the time between the first and second instants at which \(P\) hits the ground. [4]

A particle \(P\) moves with constant acceleration \((3\mathbf{i} - 2\mathbf{j}) \text{ms}^{-2}\). At time \(t = 4\) seconds, \(P\) has velocity \(6\mathbf{i} \text{ms}^{-1}\). Determine the speed of \(P\) at time \(t = 0\) seconds. [4]

A particle P lies at rest on a smooth horizontal table. A constant resultant force, F newtons, is then applied to P. As a result P moves in a straight line with constant acceleration \(\begin{bmatrix}8\\6\end{bmatrix}\) m s⁻²

1. Show that the magnitude of the acceleration of P is 10 m s⁻² [1 mark]
2. Find the speed of P after 3 seconds. [1 mark]
3. Given that \(\mathbf{F} = \begin{bmatrix}2\\1.5\end{bmatrix}\) N, find the mass of P. [2 marks]

7 A helicopter is initially at rest on the ground at the origin when it begins to accelerate in a vertical plane. Its acceleration is \(( 4.2 \mathbf { i } + 2.5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\) for the first 20 seconds of its motion. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical respectively. Assume that the helicopter moves over horizontal ground.

1. Find the height of the helicopter above the ground at the end of the 20 seconds.
2. Find the velocity of the helicopter at the end of the 20 seconds.
3. Find the speed of the helicopter when it is at a height of 180 metres above the ground.

5
An egg falls from rest a distance of 75 cm to the floor.
Neglecting air resistance, at what speed does it hit the floor?

1. In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A ball, initially at rest, is dropped from a height of 40 m above the ground.
Calculate the speed of the ball when it reaches the ground.

7

1. Find Dominic's speed at the point when the cord initially becomes taut.
   7
2. Determine whether or not Dominic enters the river and gets wet.
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3. One limitation of this model is that Dominic is not a particle.
   Explain the effect of revising this assumption on your answer to part (b). \includegraphics[max width=\textwidth, alt={}, center]{1b79a789-c003-46c9-9235-254c1d8a0501-12_2492_1721_217_150} Question number Additional page, if required.
   Write the question numbers in the left-hand margin. Question number Additional page, if required.
   Write the question numbers in the left-hand margin. Additional page, if required.
   Write the question numbers in the left-hand margin.

10 A small ball \(B\) is projected vertically upwards from a point 2 m above horizontal ground. \(B\) is projected with initial speed \(3.5 \mathrm {~ms} ^ { - 1 }\), and takes \(t\) seconds to reach the ground. Find the value of \(t\).

3. A competitor makes a dive from a high springboard into a diving pool. She leaves the springboard vertically with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) upwards. When she leaves the springboard, she is 5 m above the surface of the pool. The diver is modelled as a particle moving vertically under gravity alone and it is assumed that she does not hit the springboard as she descends. Find

1. her speed when she reaches the surface of the pool,
2. the time taken to reach the surface of the pool.
3. State two physical factors which have been ignored in the model.

2. A ball is thrown vertically upwards with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\), which is \(h\) metres above the ground. The ball moves freely under gravity until it hits the ground 5 s later.

1. Find the value of \(h\). A second ball is thrown vertically downwards with speed \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) and moves freely under gravity until it hits the ground. The first ball hits the ground with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the second ball hits the ground with speed \(\frac { 3 } { 4 } V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the value of \(w\).

1 A particle \(P\) is projected vertically upwards with speed \(11 \mathrm {~ms} ^ { - 1 }\) from a point on horizontal ground. At the same instant a particle \(Q\) is released from rest at a point \(h \mathrm {~m}\) above the ground. \(P\) and \(Q\) hit the ground at the same instant, when \(Q\) has speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the time after projection at which \(P\) hits the ground.
2. Hence find the values of \(h\) and \(V\).

A particle \(P\) is projected vertically upwards from horizontal ground with speed \(15\,\text{m}\,\text{s}^{-1}\).

1. Find the speed of \(P\) when it is 10 m above the ground. [2] At the same instant that \(P\) is projected, a second particle \(Q\) is dropped from a height of 18 m above the ground in the same vertical line as \(P\).
2. Find the height above the ground at which the two particles collide. [3]

5 Two particles, \(P\) and \(Q\), of masses \(2 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively, are held at rest in the same vertical line. The heights of \(P\) and \(Q\) above horizontal ground are 1 m and 2 m respectively. \(P\) is projected vertically upwards with speed \(2 \mathrm {~ms} ^ { - 1 }\). At the same instant, \(Q\) is released from rest.

1. Find the speed of each particle immediately before they collide.
2. It is given that immediately after the collision the downward speed of \(Q\) is \(3.5 \mathrm {~ms} ^ { - 1 }\). Find the speed of \(P\) at the instant that it reaches the ground.

The graph below shows the velocity of an object moving in a straight line over a 20 second journey. \includegraphics{figure_4}

1. Find the maximum magnitude of the acceleration of the object. [1 mark]
2. The object is at its starting position at times 0, \(t_1\) and \(t_2\) seconds. Find \(t_1\) and \(t_2\) [4 marks]

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 \(O\).

4 A particle is projected vertically upwards from a point O at \(21 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the greatest height reached by the particle. When this particle is at its highest point, a second particle is projected vertically upwards from \(O\) at \(15 \mathrm {~ms} ^ { - 1 }\).
2. Show that the particles collide 1.5 seconds later and determine the height above O at which the collision takes place.

In this question use \(g = 9.8\) m s\(^{-2}\) A ball is launched vertically upwards from the Earth's surface with velocity \(u\) m s\(^{-1}\) The ball reaches a maximum height of 15 metres. You may assume that air resistance can be ignored. Find the value of \(u\) [3 marks]

4 Two particles \(A\) and \(B\) are projected vertically upwards from horizontal ground at the same instant. The speeds of projection of \(A\) and \(B\) are \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Find

1. the difference in the heights of \(A\) and \(B\) when \(A\) is at its maximum height,
2. the height of \(A\) above the ground when \(B\) is 0.9 m above \(A\).

14 A man runs at a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight horizontal road. A woman is standing on a bridge that spans the road. At the instant that the man passes directly below the woman she throws a ball with initial speed \(u \mathrm {~ms} ^ { - 1 }\) at \(\alpha ^ { \circ }\) above the horizontal. The path of the ball is directly above the road. The man catches the ball 2.4 s after it is thrown. At the instant the man catches it, the ball is 3.6 m below the level of the point of projection.

1. Explain what it means that the ball is modelled as a particle.
2. Find the vertical component of the ball's initial velocity.
3. Find each of the following.

4 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.

4 A small ball of mass 0.4 kg is released from rest at a point 5 m above horizontal ground. At the instant the ball hits the ground it loses 12.8 J of kinetic energy and starts to move upwards.

1. Show that the greatest height above the ground that the ball reaches after hitting the ground is 1.8 m .
2. Find the time taken for the ball's motion from its release until reaching this greatest height.

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.

1. Two students observe a book of mass 0.2 kg fall vertically from rest from a shelf that is 1.5 m above the floor.

Student \(A\) suggests that the book is modelled as a particle falling freely under gravity.

1. Use student \(A\) 's model to find the time taken for the book to reach the floor. Student \(B\) suggests an improved model where the book is modelled as a particle experiencing a constant resistance to motion of magnitude \(R\) newtons. Given that the time taken for the book to reach the floor is 0.6 seconds,
2. use student \(B\) 's model to find the value of \(R\)

7 A particle moves on a smooth horizontal surface with acceleration \(( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). Initially the velocity of the particle is \(4 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find an expression for the velocity of the particle at time \(t\) seconds.
2. Find the time when the particle is travelling in the \(\mathbf { i }\) direction.
3. Show that when \(t = 4\) the speed of the particle is \(20 \mathrm {~ms} ^ { - 1 }\).

A ball, initially at rest, is dropped from a vertical height of \(h\) metres above the Earth's surface. After 4 seconds the ball's height above the Earth's surface is \(0.2h\) metres.

1. Assuming air resistance can be ignored, show that $$h = 10g$$ [3 marks]
2. Assuming air resistance cannot be ignored, explain the effect that this would have on the value of \(h\) in part (a). [1 mark]

2 An unmanned craft lands on the planet Mars. A small bolt falls from the craft onto the surface of the planet. It falls 1.5 m from rest in 0.9 s . Calculate the acceleration due to gravity on Mars.

Two particles \(A\) and \(B\) are released from rest from different starting points above a horizontal surface. \(A\) is released from a height of \(h\) metres. \(B\) is released at a time \(t\) seconds after \(A\) from a height of \(kh\) metres, where \(0 < k < 1\) Both particles land on the surface \(5\) seconds after \(A\) was released. Assuming any resistance forces may be ignored, prove that $$t = 5(1 - \sqrt{k})$$ Fully justify your answer. [5 marks]