3.02h Motion under gravity: vector form

7 A particle is projected from the origin with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. After 2 seconds the particle is at a point which is 18 m horizontally from the origin and 4 m above it.

1. Show that \(\tan \theta = \frac { 4 } { 3 }\) and find \(u\).
2. Find the horizontal range of the particle.

10 A particle is projected up a long smooth slope at a speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The slope is at an angle \(\theta\) to the horizontal where \(\sin \theta = \frac { 1 } { 25 }\). After 2 seconds it passes a mark on the slope. Find the total time taken from the moment of projection until it passes the mark again and the total distance travelled in that time. {www.cie.org.uk} after the live examination series. }

10 A small body of mass \(m\) is thrown vertically upwards with initial velocity \(u\). Resistance to motion is \(k v ^ { 2 }\) per unit mass, where the velocity is \(v\) and \(k\) is a positive constant. Find, in terms of \(u , g\) and \(k\),

1. the time taken to reach the greatest height,
2. the greatest height to which the body will rise.

12 Points \(A\) and \(B\) lie on a line of greatest slope of a plane inclined at an angle \(\alpha\) to the horizontal, with \(B\) above \(A\). A particle is projected from \(A\) with speed \(u\) at an angle \(\theta\) to the plane and subsequently strikes the plane at right angles at \(B\).

1. Show that \(2 \tan \alpha \tan \theta = 1\).
2. In either order, show that
   1. the vertical height of \(B\) above \(A\) is \(\frac { 2 u ^ { 2 } \tan ^ { 2 } \alpha } { g \left( 1 + 4 \tan ^ { 2 } \alpha \right) }\),
   2. the time of flight from \(A\) to \(B\) is \(\frac { 2 u \sec \alpha } { g \sqrt { 1 + 4 \tan ^ { 2 } \alpha } }\).

7 A building 33.8 m high stands on horizontal ground. A particle is projected horizontally from the top of the building and hits the ground 31.2 m away.

1. Find the initial speed of the particle.
2. Find the magnitude and direction of the velocity of the particle when it hits the ground.

8 A small ball is thrown vertically upwards with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), from a point 5 m above the ground. Assuming air resistance is negligible, find

1. the greatest height above the ground reached by the ball,
2. the time taken for the ball to reach the ground.

12 A particle is projected from the origin with speed \(20 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.

1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( l + \tan ^ { 2 } \alpha \right) .$$
2. Regarding the equation of the trajectory as a quadratic equation in \(\tan \alpha\), show that \(\tan \alpha\) has real values provided that $$y \leqslant 20 - \frac { x ^ { 2 } } { 80 } .$$
3. A plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The line \(l\), with equation \(y = x \tan 30 ^ { \circ }\), is a line of greatest slope in the plane. The particle is projected from the origin with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the curve \(y = 20 - \frac { x ^ { 2 } } { 80 }\), find the maximum range up this inclined plane.

14 A particle \(P\) is projected from the point \(O\), at the top of a vertical wall of height \(H\) above a horizontal plane, with initial speed \(V\) at an angle \(\alpha\) above the horizontal. At time \(t\) the coordinates of the particle are \(( x , y )\) referred to horizontal and vertical axes at \(O\).

1. Express \(x\) and \(y\) as functions of \(t\). Let \(\theta\) be the angle \(O P\) makes with the horizontal at time \(t\).
2. (a) Show that $$\tan \theta = \tan \alpha - \frac { g } { 2 V \cos \alpha } t$$ (b) Show that when the particle attains its greatest height above the point of projection, where \(O P\) makes an angle \(\beta\) with the horizontal, $$\tan \beta = \frac { 1 } { 2 } \tan \alpha .$$ (c) If the particle strikes the ground where \(O P\) makes an angle \(\beta\) below the horizontal, show that $$H = \frac { 3 V ^ { 2 } \sin ^ { 2 } \alpha } { 2 g }$$

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) and \(P\) is held with the string taut and horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt{(2ag)}\) so that it begins to move along a circular path. The string becomes slack when \(OP\) makes an angle \(\theta\) with the upward vertical through \(O\).

1. Show that \(\cos \theta = \frac{2}{3}\). [5]
2. Find the greatest height, above the horizontal through \(O\), reached by \(P\) in its subsequent motion. [4]

A particle \(P\) is projected vertically upwards with speed \(5\text{ m s}^{-1}\) from a point \(A\) which is \(2.8\text{ m}\) above horizontal ground.

1. Find the greatest height above the ground reached by \(P\). [3]
2. Find the length of time for which \(P\) is at a height of more than \(3.6\text{ m}\) above the ground. [4]

A particle \(P\) of mass \(0.3\text{ kg}\), lying on a smooth plane inclined at \(30°\) to the horizontal, is released from rest. \(P\) slides down the plane for a distance of \(2.5\text{ m}\) and then reaches a horizontal plane. There is no change in speed when \(P\) reaches the horizontal plane. A particle \(Q\) of mass \(0.2\text{ kg}\) lies at rest on the horizontal plane \(1.5\text{ m}\) from the end of the inclined plane (see diagram). \(P\) collides directly with \(Q\). \includegraphics{figure_7}

1. It is given that the horizontal plane is smooth and that, after the collision, \(P\) continues moving in the same direction, with speed \(2\text{ m s}^{-1}\). Find the speed of \(Q\) after the collision. [5]
2. It is given instead that the horizontal plane is rough and that when \(P\) and \(Q\) collide, they coalesce and move with speed \(1.2\text{ m s}^{-1}\). Find the coefficient of friction between \(P\) and the horizontal plane. [5]

Two particles \(A\) and \(B\), of masses 0.4 kg and 0.2 kg respectively, are moving down the same line of greatest slope of a smooth plane. The plane is inclined at 30° to the horizontal, and \(A\) is higher up the plane than \(B\). When the particles collide, the speeds of \(A\) and \(B\) are 3 m s\(^{-1}\) and 2 m s\(^{-1}\) respectively. In the collision between the particles, the speed of \(A\) is reduced to 2.5 m s\(^{-1}\).

1. Find the speed of \(B\) immediately after the collision. [2]

After the collision, when \(B\) has moved 1.6 m down the plane from the point of collision, it hits a barrier and returns back up the same line of greatest slope. \(B\) hits the barrier 0.4 s after the collision, and when it hits the barrier, its speed is reduced by 90%. The two particles collide again 0.44 s after their previous collision, and they then coalesce on impact.

1. Show that the speed of \(B\) immediately after it hits the barrier is 0.5 m s\(^{-1}\). Hence find the speed of the combined particle immediately after the second collision between \(A\) and \(B\). [7]

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]

A particle is projected vertically upwards from horizontal ground. The speed of the particle 2 seconds after it is projected is \(5\) m s\(^{-1}\) and it is travelling downwards.

1. Find the speed of projection of the particle. [2]
2. Find the distance travelled by the particle between the two times at which its speed is \(10\) m s\(^{-1}\). [2]

A particle \(A\) of mass 0.5 kg is projected vertically upwards from horizontal ground with speed 25 m s\(^{-1}\).

1. Find the speed of \(A\) when it reaches a height of 20 m above the ground. [2]

When \(A\) reaches a height of 20 m, it collides with a particle \(B\) of mass 0.3 kg which is moving downwards in the same vertical line as \(A\) with speed 32.5 m s\(^{-1}\). In the collision between the two particles, \(B\) is brought to instantaneous rest.

1. Show that the velocity of \(A\) immediately after the collision is 4.5 m s\(^{-1}\) downwards. [2]
2. Find the time interval between \(A\) and \(B\) reaching the ground. You should assume that \(A\) does not bounce when it reaches the ground. [4]

A particle, \(A\), is projected vertically upwards from a point \(O\) with a speed of \(80 \text{ ms}^{-1}\). One second later a second particle, \(B\), with the same mass as \(A\), is projected vertically upwards from \(O\) with a speed of \(100 \text{ ms}^{-1}\). At time \(T\) s after the first particle is projected, the two particles collide and coalesce to form a particle \(C\).

1. Show that \(T = 3.5\). [4]
2. Find the height above \(O\) at which the particles collide. [1]
3. Find the time from \(A\) being projected until \(C\) returns to \(O\). [5]

\includegraphics{figure_7} Two particles, \(A\) and \(B\), of masses 3 kg and 5 kg respectively, are connected by a light inextensible string that passes over a fixed smooth pulley. The particles are held with the string taut and its straight parts vertical. Particle \(A\) is 1 m above a horizontal plane, and particle \(B\) is 2 m above the plane (see diagram). The particles are released from rest. In the subsequent motion, \(A\) does not reach the pulley, and after \(B\) reaches the plane it remains in contact with the plane.

1. Find the tension in the string and the time taken for \(B\) to reach the plane. [6]
2. Find the time for which \(A\) is at least 3.25 m above the plane. [4]

\includegraphics{figure_6} A particle is projected vertically upward from ground level with speed \(u\) m s\(^{-1}\). The particle moves under gravity alone.

1. Find an expression for the maximum height reached by the particle. [3]

\includegraphics{figure_6b} The diagram shows a velocity-time graph for the motion of the particle.

1. Use the graph to find the value of \(u\). [2]
2. Find the time taken for the particle to return to ground level. [3]

A particle \(P\) is projected vertically upwards with speed \(24 \text{ m s}^{-1}\) from a point \(5 \text{ m}\) above ground level. Find the time from projection until \(P\) reaches the ground. [3]

A small ball is projected vertically downwards with speed \(5\text{ m s}^{-1}\) from a point \(A\) at a height of \(7.2\text{ m}\) above horizontal ground. The ball hits the ground with speed \(V\text{ m s}^{-1}\) and rebounds vertically upwards with speed \(\frac{1}{2}V\text{ m s}^{-1}\). The highest point the ball reaches after rebounding is \(B\). Find \(V\) and hence find the total time taken for the ball to reach the ground from \(A\) and rebound to \(B\). [5]

A particle is projected vertically upwards with speed \(30\) m s\(^{-1}\) from a point on horizontal ground.

1. Show that the maximum height above the ground reached by the particle is \(45\) m. [2]
2. Find the time that it takes for the particle to reach a height of \(33.75\) m above the ground for the first time. Find also the speed of the particle at this time. [4]

A particle is projected vertically upwards from a point \(O\) with initial speed \(12.5 \text{ m s}^{-1}\). At the same instant another particle is released from rest at a point 10 m vertically above \(O\). Find the height above \(O\) at which the particles meet. [5]

A particle \(P\) is projected vertically upwards from horizontal ground with speed 12 m s\(^{-1}\).

1. Find the time taken for \(P\) to return to the ground. [2]

The time in seconds after \(P\) is projected is denoted by \(t\). When \(t = 1\), a second particle \(Q\) is projected vertically upwards with speed 10 m s\(^{-1}\) from a point which is 5 m above the ground. Particles \(P\) and \(Q\) move in different vertical lines.

1. Find the set of values of \(t\) for which the two particles are moving in the same direction. [4]

A particle of mass 0.4 kg is released from rest at a height of 1.8 m above the surface of the water in a tank. There is no instantaneous change of speed when the particle enters the water. The water exerts an upward force of 5.6 N on the particle when it is in the water.

1. Find the velocity of the particle at the instant when it reaches the surface of the water. [2]

1. Find the time that it takes from the instant when the particle enters the water until it comes to instantaneous rest in the water. You may assume that the tank is deep enough so that the particle does not reach the bottom of the tank. [4]

1. Sketch a velocity-time graph for the motion of the particle from the instant at which it is released until it comes to instantaneous rest in the water. [3]

Two particles \(A\) and \(B\) move in the same vertical line. Particle \(A\) is projected vertically upwards from the ground with speed 20 m s\(^{-1}\). One second later particle \(B\) is dropped from rest from a height of 40 m.

1. Find the height above the ground at which the two particles collide. [4]
2. Find the difference in the speeds of the two particles at the instant when the collision occurs. [3]