3.02h Motion under gravity: vector form

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 ball of mass 0.2 kg is projected vertically downwards with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point A which is 2.5 m above horizontal ground. The ball hits the ground. Immediately after hitting the ground, the ball rebounds vertically with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball receives an impulse of magnitude 7 Ns in its impact with the ground. By modelling the ball as a particle and ignoring air resistance, find

1. the value of \(U\). After hitting the ground, the ball moves vertically upwards and passes through a point \(B\) which is 1 m above the ground.
2. Find the time between the instant when the ball hits the ground and the instant when the ball first passes through \(B\).
3. Sketch a velocity-time graph for the motion of the ball from when it was projected from \(A\) to when it first passes through \(B\). (You need not make any further calculations to draw this sketch.)

6. \begin{figure}[h] \end{figure} A small ball \(P\) is projected with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A 10 \mathrm {~m}\) above horizontal ground. The angle of projection is \(55 ^ { \circ }\) above the horizontal. The ball moves freely under gravity and hits the ground at the point \(B\), as shown in Figure 3. Find

1. the speed of \(P\) as it hits the ground at \(B\),
2. the direction of motion of \(P\) as it hits the ground at \(B\),
3. the time taken for \(P\) to move from \(A\) to \(B\).

7. A particle is projected from a point \(O\) with speed \(U\) at an angle of elevation \(\alpha\) to the horizontal and moves freely under gravity. When the particle has moved a horizontal distance \(x\), its height above \(O\) is \(y\).

1. Show that $$y = x \tan \alpha - \frac { g x ^ { 2 } \left( 1 + \tan ^ { 2 } \alpha \right) } { 2 U ^ { 2 } }$$ \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{54112b4a-3727-4e5b-97e5-4291e7172438-22_330_857_632_548} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} A small stone is projected horizontally with speed \(U\) from a point \(C\) at the top of a vertical cliff \(A C\) so as to hit a fixed target \(B\) on the horizontal ground. The point \(C\) is a height \(h\) above the ground, as shown in Figure 3. The time of flight of the stone from \(C\) to \(B\) is \(T\), and the stone is modelled as a particle moving freely under gravity.
2. Find, in terms of \(U , g\) and \(T\), the speed of the stone as it hits the target at \(B\). It is found that, using the same initial speed \(U\), the target can also be hit by projecting the stone from \(C\) at an angle \(\alpha\) above the horizontal. The stone is again modelled as a particle moving freely under gravity and the distance \(A B = d\).
3. Using the result in part (a), or otherwise, show that $$d = \frac { 1 } { 2 } g T ^ { 2 } \tan \alpha$$

8. \begin{figure}[h] \end{figure} A rough ramp \(A B\) is fixed to horizontal ground at \(A\). The ramp is inclined at \(20 ^ { \circ }\) to the ground. The line \(A B\) is a line of greatest slope of the ramp and \(A B = 6 \mathrm {~m}\). The point \(B\) is at the top of the ramp, as shown in Figure 3. A particle \(P\) of mass 3 kg is projected with speed \(15 \mathrm {~ms} ^ { - 1 }\) from \(A\) towards \(B\). At the instant \(P\) reaches the point \(B\) the speed of \(P\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The force due to friction is modelled as a constant force of magnitude \(F\) newtons.

1. Use the work-energy principle to find the value of \(F\). After leaving the ramp at \(B\), the particle \(P\) moves freely under gravity until it hits the horizontal ground at the point \(C\). The speed of \(P\) as it hits the ground at \(C\) is \(w \mathrm {~ms} ^ { - 1 }\). Find
2. 1. the value of \(w\),
   2. the direction of motion of \(P\) as it hits the ground at \(C\),
3. the greatest height of \(P\) above the ground as \(P\) moves from \(A\) to \(C\).

7. \begin{figure}[h] \end{figure} At time \(t = 0\) a particle \(P\) is projected from a fixed point \(A\) on horizontal ground. The particle is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to the ground. The particle moves freely under gravity. At time \(t = 3\) seconds, \(P\) is passing through the point \(B\) with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving downwards at an angle \(\beta\) to the horizontal, as shown in Figure 5.

1. By considering energy, find the height of \(B\) above the ground.
2. Find the size of angle \(\alpha\).
3. Find the size of angle \(\beta\).
4. Find the least speed of \(P\) as \(P\) travels from \(A\) to \(B\). As \(P\) travels from \(A\) to \(B\), the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is such that \(v \leqslant 15\) for an interval of \(T\) seconds.
5. Find the value of \(T\).
   \section*{\textbackslash section*\{Question 7 continued\}}

7. \begin{figure}[h] \end{figure} The fixed point \(A\) is 20 m vertically above the point \(O\) which is on horizontal ground. At time \(t = 0\), a particle \(P\) is projected from \(A\) with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta ^ { \circ }\) above the horizontal. The particle moves freely under gravity. At time \(t = 5\) seconds, \(P\) strikes the ground at the point \(B\), where \(O B = 40 \mathrm {~m}\), as shown in Figure 4.

1. By considering energy, find the speed of \(P\) as it hits the ground at \(B\).
2. Find the least speed of \(P\) as it moves from \(A\) to \(B\).
3. Find the length of time for which the speed of \(P\) is more than \(10 \mathrm {~ms} ^ { - 1 }\).

8. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a block \(A\) of mass 3 kg . Block \(A\) is held at rest on a smooth fixed plane. The plane is inclined at \(40 ^ { \circ }\) to the horizontal ground. The string lies along a line of greatest slope of the plane and passes over a small smooth pulley which is fixed at the top of the plane. The other end of the string is attached to a block \(B\) of mass 5 kg . Block \(B\) hangs freely at rest below the pulley, as shown in Figure 4. The system is released from rest with the string taut. By modelling the two blocks as particles,

1. find the tension in the string as \(B\) descends. After falling for 1.5 s , block \(B\) hits the ground and is immediately brought to rest. In its subsequent motion, \(A\) does not reach the pulley.
2. Find the speed of \(B\) at the instant it hits the ground.
3. Find the total distance moved up the plane by \(A\) before it comes to instantaneous rest. \includegraphics[max width=\textwidth, alt={}, center]{04b73f81-3316-4f26-ad98-a7be3a4b738f-28_97_141_2519_1804} \includegraphics[max width=\textwidth, alt={}, center]{04b73f81-3316-4f26-ad98-a7be3a4b738f-28_125_161_2624_1779}

3. A particle, \(P\), is projected vertically upwards with speed \(U\) from a fixed point \(O\). At the instant when \(P\) reaches its greatest height \(H\) above \(O\), a second particle, \(Q\), is projected with speed \(\frac { 1 } { 2 } U\) vertically upwards from \(O\).

1. Find \(H\) in terms of \(U\) and \(g\).
2. Find, in terms of \(U\) and \(g\), the time between the instant when \(Q\) is projected and the instant when the two particles collide.
3. Find where the two particles collide. DO NOT WRITEIN THIS AREA \includegraphics[max width=\textwidth, alt={}, center]{916543cb-14f7-486c-ba3c-eda9be134045-08_2666_99_107_1957}

1. A small stone is projected vertically upwards with speed \(20 \mathrm {~ms} ^ { - 1 }\) from a point \(O\) which is 5 m above horizontal ground. The stone is modelled as a particle moving freely under gravity.

Find

1. the speed of the stone at the instant when it is 2 m above the ground,
2. the total time between the instant when the stone is projected from \(O\) and the instant when it first strikes the ground.

7. A helicopter is hovering at rest above horizontal ground at the point \(H\). A parachutist steps out of the helicopter and immediately falls vertically and freely under gravity from rest for 2.5 s . His parachute then opens and causes him to immediately decelerate at a constant rate of \(3.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for \(T\) seconds ( \(T < 6\) ), until his speed is reduced to \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He then moves with this constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) until he hits the ground. While he is decelerating, he falls a distance of 73.75 m . The total time between the instant when he leaves \(H\) and the instant when he hits the ground is 20 s . The parachutist is modelled as a particle.

1. Find the speed of the parachutist at the instant when his parachute opens.
2. Sketch a speed-time graph for the motion of the parachutist from the instant when he leaves \(H\) to the instant when he hits the ground.
3. Find the value of \(T\).
4. Find, to the nearest metre, the height of the point \(H\) above the ground.
   7. A helicopter is hovering at rest above horizontal ground at the point \(H\). A parachutist steps

4. At time \(t = 0\), a small ball is projected vertically upwards from a point \(A\) which is 24.5 m above the ground. The ball first comes to instantaneous rest at the point \(B\), where \(A B = 19.6 \mathrm {~m}\) and first hits the ground at time \(t = T\) seconds. The ball is modelled as a particle moving freely under gravity.

1. Find the value of \(T\).
2. Sketch a speed-time graph for the motion of the ball from \(t = 0\) to \(t = T\) seconds.
   (No further calculations are needed in order to draw this sketch.)

1. A particle is projected vertically upwards from a point \(A\) with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

The point \(A\) is 2.5 m vertically above the point \(B\).
Point \(B\) lies on horizontal ground.
The particle moves freely under gravity until it hits the ground at \(B\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) After hitting the ground the particle does not rebound.

1. Find the value of \(V\).
2. Find the time taken for the particle to reach \(B\). The point \(C\) is 10 m vertically above \(A\).
3. Find the length of time for which the particle is above \(C\).
4. Sketch a speed-time graph for the motion of the particle from projection to the instant that it reaches \(B\). (No further calculations are required.)

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 small stone is released from rest from a point \(A\) which is at height \(h\) metres above horizontal ground. Exactly one second later another small stone is projected with speed \(19.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards from a point \(B\), which is also at height \(h\) metres above the horizontal ground. The motion of each stone is modelled as that of a particle moving freely under gravity. The two stones hit the ground at the same time.

Find the value of \(h\).

2. A small ball is thrown vertically upwards with speed \(14.7 \mathrm {~ms} ^ { - 1 }\) from a point that is 19.6 m above horizontal ground. The ball is modelled as a particle moving freely under gravity. Find

1. the total time from when the ball is thrown to when it first hits the ground,
2. the speed of the ball immediately before it first hits the ground,
3. the total distance travelled by the ball from when it is thrown to when it first hits the ground.
4. Sketch a velocity-time graph for the motion of the ball from when it is thrown to when it first hits the ground. State the coordinates of the start point and the coordinates of the end point of your graph.
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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 \(R\)

1. A parachute is used to deliver a box of supplies. The parachute is attached to the box.

• the parachute and box are dropped from rest from a helicopter that is hovering at a height of 520 m above the ground
• the parachute and box fall vertically and freely under gravity for 5 seconds, then the parachute opens
• from the instant the parachute opens, it provides a resistance to motion of magnitude 3200 N
• the parachute and box continue to fall vertically downwards after the parachute opens
• the parachute and box are modelled throughout the motion as a particle \(P\) of mass 250 kg
  1. Find the distance fallen by \(P\) in the first 5 seconds.
  2. Find the speed with which \(P\) lands on the ground.
  3. Find the total time from the instant when \(P\) is dropped from the helicopter to the instant when \(P\) lands on the ground.
  4. Sketch a speed-time graph for the motion of \(P\) from the instant when \(P\) is dropped from the helicopter to the instant when \(P\) lands on the ground.

1. At time \(t = 0\), a stone is thrown vertically upwards with speed \(19.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) which is \(h\) metres above horizontal ground. At time \(t = 3 \mathrm {~s}\), another stone is released from rest from a point \(B\) which is also \(h\) metres above the same horizontal ground. Both stones hit the ground at time \(t = T\) seconds. The motion of each stone is modelled as that of a particle moving freely under gravity.

Find

1. the value of \(T\),
2. the value of \(h\).
   

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6. \begin{figure}[h] \end{figure} A small ball is thrown vertically upwards at time \(t = 0\) from a point \(A\) which is above horizontal ground. The ball hits the ground 7 s later. The ball is modelled as a particle moving freely under gravity.
The velocity-time graph shown in Figure 3 represents the motion of the ball for \(0 \leqslant t \leqslant 7\)

1. Find the speed with which the ball is thrown.
2. Find the height of \(A\) above the ground.

3. A block \(A\) of mass 9 kg is released from rest from a point \(P\) which is a height \(h\) metres above horizontal soft ground. The block falls and strikes another block \(B\) of mass 1.5 kg which is on the ground vertically below \(P\). The speed of \(A\) immediately before it strikes \(B\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The blocks are modelled as particles.

1. Find the value of \(h\). Immediately after the impact the blocks move downwards together with the same speed and both come to rest after sinking a vertical distance of 12 cm into the ground. Assuming that the resistance offered by the ground has constant magnitude \(R\) newtons,
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