Two projectiles meeting - 2D flight

5 \includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-4_547_933_269_607} Particles \(A\) and \(B\) are projected simultaneously from the top \(T\) of a vertical tower, and move in the same vertical plane. \(T\) is 7.2 m above horizontal ground. \(A\) is projected horizontally with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) is projected at an angle of \(60 ^ { \circ }\) above the horizontal with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 } . A\) and \(B\) move away from each other (see diagram).

1. Find the time taken for \(A\) to reach the ground. At the instant when \(A\) hits the ground,
2. show that \(B\) is approximately 5.2 m above the ground,
3. find the distance \(A B\).

3 \includegraphics[max width=\textwidth, alt={}, center]{fe5c198d-5d05-4241-98f5-894ba92f7afe-3_408_1164_248_493} A particle \(P\) is released from rest at a point \(A\) which is 7 m above horizontal ground. At the same instant that \(P\) is released a particle \(Q\) is projected from a point \(O\) on the ground. The horizontal distance of \(O\) from \(A\) is 24 m . Particle \(Q\) moves in the vertical plane containing \(O\) and \(A\), with initial speed \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and initial direction making an angle \(\theta\) above the horizontal, where \(\tan \theta = \frac { 7 } { 24 }\) (see diagram). Show that the particles collide.

6. \begin{figure}[h] \end{figure} The points \(A\) and \(B\) lie 40 m apart on horizontal ground. At time \(t = 0\) the particles \(P\) and \(Q\) are projected in the vertical plane containing \(A B\) and move freely under gravity. Particle \(P\) is projected from \(A\) with speed \(30 \mathrm {~ms} ^ { - 1 }\) at \(60 ^ { \circ }\) to \(A B\) and particle \(Q\) is projected from \(B\) with speed \(q \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at angle \(\theta\) to \(B A\), as shown in Figure 4. At \(t = 2\) seconds, \(P\) and \(Q\) collide.

1. Find
   1. the size of angle \(\theta\),
   2. the value of \(q\).
2. Find the speed of \(P\) at the instant before it collides with \(Q\).

6 \includegraphics[max width=\textwidth, alt={}, center]{8e1225a2-cb98-4b71-a4af-0150f093f852-3_698_1047_1297_550} A particle \(P\) is projected with speed \(V _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta _ { 1 }\) from a point \(O\) on horizontal ground. When \(P\) is vertically above a point \(A\) on the ground its height is 250 m and its velocity components are \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) horizontally and \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically upwards (see diagram).

1. Show that \(V _ { 1 } = 86.0\) and \(\theta _ { 1 } = 62.3 ^ { \circ }\), correct to 3 significant figures. At the instant when \(P\) is vertically above \(A\), a second particle \(Q\) is projected from \(O\) with speed \(V _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta _ { 2 } . P\) and \(Q\) hit the ground at the same time and at the same place.
2. Calculate the total time of flight of \(P\) and the total time of flight of \(Q\).
3. Calculate the range of the particles and hence calculate \(V _ { 2 }\) and \(\theta _ { 2 }\).

7 A particle \(P\) is projected horizontally with speed \(15 \mathrm {~ms} ^ { - 1 }\) from the top of a vertical cliff. At the same instant a particle \(Q\) is projected from the bottom of the cliff, with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. \(P\) and \(Q\) move in the same vertical plane. The height of the cliff is 60 m and the ground at the bottom of the cliff is horizontal.

1. Given that the particles hit the ground simultaneously, find the value of \(\theta\) and find also the distance between the points of impact with the ground.
2. Given instead that the particles collide, find the value of \(\theta\), and determine whether \(Q\) is rising or falling immediately before this collision.

5. \begin{figure}[h] \end{figure} The points \(A\) and \(B\) lie 50 m apart on horizontal ground.
At time \(t = 0\) two small balls, \(P\) and \(Q\), are projected in the vertical plane containing \(A B\).
Ball \(P\) is projected from \(A\) with speed \(20 \mathrm {~ms} ^ { - 1 }\) at \(30 ^ { \circ }\) to \(A B\).
Ball \(Q\) is projected from \(B\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at angle \(\theta\) to \(B A\), as shown in Figure 3.
At time \(t = 2\) seconds, \(P\) and \(Q\) collide.
Until they collide, the balls are modelled as particles moving freely under gravity.

1. Find the velocity of \(P\) at the instant before it collides with \(Q\).
2. Find
   1. the size of angle \(\theta\),
   2. the value of \(u\).
3. State one limitation of the model, other than air resistance, that could affect the accuracy of your answers.

8 In this question the value of \(\boldsymbol { g \) should be taken as \(\mathbf { 1 0 } \mathbf { m ~ s } ^ { \mathbf { - 2 } }\).} As shown in Fig. 8, particles A and B are projected towards one another. Each particle has an initial speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically and \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) horizontally. Initially A and B are 70 m apart horizontally and B is 15 m higher than A . Both particles are projected over horizontal ground. \begin{figure}[h] \end{figure}

1. Show that, \(t\) seconds after projection, the height in metres of each particle above its point of projection is \(10 t - 5 t ^ { 2 }\).
2. Calculate the horizontal range of A . Deduce that A hits the horizontal ground between the initial positions of A and B .
3. Calculate the horizontal distance travelled by B before reaching the ground.
4. Show that the paths of the particles cross but that the particles do not collide if they are projected at the same time. In fact, particle A is projected 2 seconds after particle B .
5. Verify that the particles collide 0.75 seconds after A is projected.

2 In this question the value of \(g\) should be taken as \(10 \mathrm {~m \mathrm {~s} ^ { 2 }\).} As shown in Fig. 8, particles A and B are projected towards one another. Each particle has an initial speed of \(10 \mathrm {~m} \mathrm {~s} ^ { 1 }\) vertically and \(20 \mathrm {~m} \mathrm {~s} { } ^ { 1 }\) horizontally. Initially A and B are 70 m apart horizontally and B is 15 m higher than A . Both particles are projected over horizontal ground. \begin{figure}[h] \end{figure}

1. Show that, \(t\) seconds after projection, the height in metres of each particle above its point of projection is \(10 t - 5 t ^ { 2 }\).
2. Calculate the horizontal range of A . Deduce that A hits the horizontal ground between the initial positions of A and B .
3. Calculate the horizontal distance travelled by B before reaching the ground.
4. Show that the paths of the particles cross but that the particles do not collide if they are projected at the same time. In fact, particle A is projected 2 seconds after particle B .
5. Verify that the particles collide 0.75 seconds after A is projected.

A particle \(P\) is projected with speed \(V\text{ m s}^{-1}\) at an angle of \(60°\) above the horizontal from a point \(O\). At the instant \(1\text{ s}\) later a particle \(Q\) is projected from \(O\) with the same initial speed at an angle of \(45°\) above the horizontal. The two particles collide when \(Q\) has been in motion for \(t\text{ s}\).

1. Show that \(t = 2.414\), correct to 3 decimal places. [3]
2. Find the value of \(V\). [4]

The collision occurs after \(P\) has passed through the highest point of its trajectory.

1. Calculate the vertical distance of \(P\) below its greatest height when \(P\) and \(Q\) collide. [4]

A particle \(P\) is projected with speed \(V\,\text{m s}^{-1}\) at an angle of \(60°\) above the horizontal from a point \(O\). At the instant \(1\,\text{s}\) later a particle \(Q\) is projected from \(O\) with the same initial speed at an angle of \(45°\) above the horizontal. The two particles collide when \(Q\) has been in motion for \(t\,\text{s}\).

1. Show that \(t = 2.414\), correct to \(3\) decimal places. [3]
2. Find the value of \(V\). [4]

The collision occurs after \(P\) has passed through the highest point of its trajectory.

1. Calculate the vertical distance of \(P\) below its greatest height when \(P\) and \(Q\) collide. [4]

Particles \(P\) and \(Q\) are projected in the same vertical plane from a point \(O\) at the top of a cliff. The height of the cliff exceeds 50 m. Both particles move freely under gravity. Particle \(P\) is projected with speed \(\frac{35}{2} \text{ m s}^{-1}\) at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac{4}{3}\). Particle \(Q\) is projected with speed \(u \text{ m s}^{-1}\) at an angle \(\beta\) above the horizontal, where \(\tan \beta = \frac{1}{2}\). Particle \(Q\) is projected one second after the projection of particle \(P\). The particles collide \(T\) s after the projection of particle \(Q\).

1. Write down expressions, in terms of \(T\), for the horizontal displacements of \(P\) and \(Q\) from \(O\) when they collide and hence show that \(4uT = 21\sqrt{5(T + 1)}\). [4]
2. Find the value of \(T\). [4]
3. Find the horizontal and vertical displacements of the particles from \(O\) when they collide. [3]

The points \(O\) and \(P\) are on a horizontal plane, a distance \(8\) m apart. A ball is thrown from \(O\) with speed \(u\) m s\(^{-1}\) at an angle \(\theta\) above the horizontal, where \(\tan \theta = \frac{3}{4}\). At the same instant, a model aircraft is launched with speed \(5\) m s\(^{-1}\) parallel to the horizontal plane from a point \(4\) m vertically above \(P\). The model aircraft moves in the same vertical plane as the ball and in the same horizontal direction as the ball. The model aircraft moves horizontally with a constant speed of \(5\) m s\(^{-1}\). After \(T\) s, the ball and the model aircraft collide.

1. Find the value of \(T\). [6]
2. Find the direction in which the ball is moving immediately before the collision. [3]

A particle \(P\) is projected with speed \(u\text{ ms}^{-1}\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. During its flight \(P\) passes through the point which is a horizontal distance \(3a\) from \(O\) and a vertical distance \(\frac{3}{8}a\) above the horizontal plane. It is given that \(\tan\theta = \frac{1}{3}\).

1. Show that \(u^2 = 8ag\). [2]

A particle \(Q\) is projected with speed \(V\text{ ms}^{-1}\) at an angle \(\alpha\) above the horizontal from \(O\) at the instant when \(P\) is at its highest point. Particles \(P\) and \(Q\) both land at the same point on the horizontal plane at the same time.

1. Find \(V\) in terms of \(a\) and \(g\). [7]

A particle \(P\) is projected from a point \(O\) on horizontal ground with speed \(u\) at an angle \(\theta\) above the horizontal, where \(\tan \theta = \frac{1}{3}\). The particle \(P\) moves freely under gravity and passes through the point with coordinates \((3a, \frac{4}{5}a)\) relative to horizontal and vertical axes through \(O\) in the plane of the motion.

1. Use the equation of the trajectory to show that \(u^2 = 25ag\). [2]
2. Express \(V^2\) in the form \(kag\), where \(k\) is a rational number. [6]

At the instant when \(P\) is moving horizontally, a particle \(Q\) is projected from \(O\) with speed \(V\) at an angle \(\alpha\) above the horizontal. The particles \(P\) and \(Q\) reach the ground at the same point and at the same time.

A vertical cliff is 73.5 m high. Two stones \(A\) and \(B\) are projected simultaneously. Stone \(A\) is projected horizontally from the top of the cliff with speed 28 m s\(^{-1}\). Stone \(B\) is projected from the bottom of the cliff with speed 35 m s\(^{-1}\) at an angle \(\alpha\) above the horizontal. The stones move freely under gravity in the same vertical plane and collide in mid-air. By considering the horizontal motion of each stone,

1. prove that \(\cos \alpha = \frac{4}{5}\). [4]

1. Find the time which elapses between the instant when the stones are projected and the instant when they collide. [4]

A particle \(P\) is projected with speed \(32 \text{ m s}^{-1}\) at an angle of elevation \(\alpha\), where \(\sin \alpha = \frac{3}{4}\), from a point \(A\) on horizontal ground. At the same instant a particle \(Q\) is projected with speed \(20 \text{ m s}^{-1}\) at an angle of elevation \(\beta\), where \(\sin \beta = \frac{24}{25}\), from a point \(B\) on the same horizontal ground. The particles move freely under gravity in the same vertical plane and collide with each other at the point \(C\) at the instant when they are travelling horizontally (see diagram).

1. Calculate the height of \(C\) above the ground and the distance \(AB\). [4]

Immediately after the collision \(P\) falls vertically. \(P\) hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height 5 m above the ground.

1. Given that the mass of \(P\) is 3 kg, find the magnitude and direction of the impulse exerted on \(P\) by the ground. [4]

The coefficient of restitution between the two particles is \(\frac{1}{2}\).

1. Find the distance of \(Q\) from \(C\) at the instant when \(Q\) is travelling in a direction of \(25°\) below the horizontal. [9]

Points \(A\) and \(B\) lie on horizontal ground. At time \(t = 0\) seconds, an object \(P\) is projected from \(A\) towards \(B\) such that \(AB\) is the range of \(P\). The speed of projection is \(24 \cdot 5\) ms\(^{-1}\) in a direction which is 30° above the horizontal.

1. Calculate the range \(AB\) of the object \(P\). [5]

At time \(t = 1\) second, another object \(Q\) is projected from \(B\) towards \(A\) with the same speed of projection \(24 \cdot 5\) ms\(^{-1}\) and in a direction which is also 30° above the horizontal.

1. Determine the height above the ground at which \(P\) and \(Q\) collide. [5]

In this question you must show detailed reasoning. \includegraphics{figure_11} A football \(P\) is kicked with speed \(25\,\text{m}\,\text{s}^{-1}\) at an angle of elevation \(\alpha\) from a point \(A\) on horizontal ground. At the same instant a second football \(Q\) is kicked with speed \(15\,\text{m}\,\text{s}^{-1}\) at an angle of elevation \(2\alpha\) from a point \(B\) on the same horizontal ground, where \(AB = 72\) m. The footballs are modelled as particles moving freely under gravity in the same vertical plane and they collide with each other at the point \(C\) (see diagram).

1. Calculate the height of \(C\) above the ground. [7]
2. Find the direction of motion of \(P\) at the moment of impact. [4]
3. Suggest one improvement that could be made to the model. [1]