3.02i Projectile motion: constant acceleration model

A particle \(P\) is projected with speed \(u\) at an angle \(\alpha\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a subsequent time \(t\) are denoted by \(x\) and \(y\) respectively.

1. Derive the equation of the trajectory of \(P\) in the form $$y = x\tan\alpha - \frac{gx^2}{2u^2}\sec^2\alpha.$$ [3]

The point \(Q\) is the highest point on the trajectory of \(P\) in the case where \(\alpha = 45°\).

1. Show that the \(x\)-coordinate of \(Q\) is \(\frac{u^2}{2g}\). [3]
2. Find the other value of \(\alpha\) for which \(P\) would pass through the point \(Q\). [4]

A particle \(P\) is projected from a point \(O\) on a horizontal plane and moves freely under gravity. Its initial speed is \(u\) ms\(^{-1}\) and its angle of projection is \(\sin^{-1}(\frac{3}{5})\) above the horizontal. At time 8 s after projection, \(P\) is at the point \(A\). At time 32 s after projection, \(P\) is at the point \(B\). The direction of motion of \(P\) at \(B\) is perpendicular to its direction of motion at \(A\). Find the value of \(u\). [7]

A particle is projected with speed \(u\) at an angle \(\alpha\) above the horizontal from a point \(O\) on a horizontal plane. The particle moves freely under gravity.

1. Write down the horizontal and vertical components of the velocity of the particle at time \(T\) after projection. [2] At time \(T\) after projection, the direction of motion of the particle is perpendicular to the direction of projection.
2. Express \(T\) in terms of \(u\), \(g\) and \(\alpha\). [2]
3. Deduce that \(T > \frac{u}{g}\). [1]

One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(m\). The particle \(P\) is held vertically below \(O\) with the string taut and then projected horizontally. When the string makes an angle of \(60°\) with the upward vertical, \(P\) becomes detached from the string. In its subsequent motion, \(P\) passes through the point \(A\) which is a distance \(a\) vertically above \(O\).

1. The speed of \(P\) when it becomes detached from the string is \(V\). Use the equation of the trajectory of a projectile to find \(V\) in terms of \(a\) and \(g\). [4]
2. Find, in terms of \(m\) and \(g\), the tension in the string immediately after \(P\) is initially projected horizontally. [4]

A particle \(P\) is projected with speed \(u\) m s\(^{-1}\) at an angle of \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a subsequent time \(t\) s are denoted by \(x\) m and \(y\) m respectively.

1. Show that the equation of the trajectory is given by $$y = x \tan \theta - \frac{gx^2}{2u^2}(1 + \tan^2 \theta).$$ [4]

In the subsequent motion \(P\) passes through the point with coordinates \((30, 20)\).

1. Given that one possible value of \(\tan \theta\) is \(\frac{4}{3}\), find the other possible value of \(\tan \theta\). [5]

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 with speed \(u\) at an angle \(\alpha\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a subsequent time \(t\) are denoted by \(x\) and \(y\) respectively.

1. Derive the equation of the trajectory of \(P\) in the form $$y = x \tan \alpha - \frac{gx^2}{2u^2} \sec^2 \alpha.$$ [3]

During its flight, \(P\) must clear an obstacle of height \(h\) m that is at a horizontal distance of \(32\) m from the point of projection. When \(u = 40\sqrt{2}\) m s\(^{-1}\), \(P\) just clears the obstacle. When \(u = 40\) m s\(^{-1}\), \(P\) only achieves \(80\%\) of the height required to clear the obstacle.

1. Find the two possible values of \(h\). [6]

A particle \(P\) is projected with speed \(u\text{ms}^{-1}\) at an angle \(\tan^{-1}2\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. When \(P\) has travelled a distance \(56\text{m}\) horizontally from \(O\), it is at a vertical height \(H\text{m}\) above the plane. When \(P\) has travelled a distance \(84\text{m}\) horizontally from \(O\), it is at a vertical height \(\frac{1}{2}H\text{m}\) above the plane. Find, in either order, the value of \(u\) and the value of \(H\). [5]

A particle \(P\) is projected with speed \(u\) at an angle \(\tan^{-1}\left(\frac{4}{3}\right)\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. When \(P\) is moving horizontally, it strikes a smooth inclined plane at the point \(A\). This plane is inclined to the horizontal at an angle \(\alpha\), and the line of greatest slope through \(A\) lies in the vertical plane through \(O\) and \(A\). As a result of the impact, \(P\) moves vertically upwards. The coefficient of restitution between \(P\) and the inclined plane is \(e\).

1. Show that \(e\tan^2\alpha = 1\). [4]

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 particle \(P\) is projected with speed \(u \text{ m s}^{-1}\) at an angle \(\tan^{-1} 2\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. When \(P\) has travelled a distance \(56 \text{ m}\) horizontally from \(O\), it is at a vertical height \(H \text{ m}\) above the plane. When \(P\) has travelled a distance \(84 \text{ m}\) horizontally from \(O\), it is at a vertical height \(\frac{1}{2}H \text{ m}\) above the plane. Find, in either order, the value of \(u\) and the value of \(H\). [5]

A particle \(P\) is projected with speed \(u\) at an angle \(\tan^{-1}\left(\frac{4}{3}\right)\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. When \(P\) is moving horizontally, it strikes a smooth inclined plane at the point \(A\). This plane is inclined to the horizontal at an angle \(\alpha\), and the line of greatest slope through \(A\) lies in the vertical plane through \(O\) and \(A\). As a result of the impact, \(P\) moves vertically upwards. The coefficient of restitution between \(P\) and the inclined plane is \(e\).

1. Show that \(e \tan^2 \alpha = 1\). [4]

A particle \(P\) is projected with speed \(u\) at an angle \(\alpha\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a time \(t\) are denoted by \(x\) and \(y\) respectively.

1. Derive the equation of the trajectory of \(P\) in the form $$y = x \tan \alpha - \frac{gx^2}{2u^2} \sec^2 \alpha.$$ [3]
2. The greatest height of \(P\) above the plane is denoted by \(H\). When \(P\) is at a height of \(\frac{3}{4}H\), it is travelling at a horizontal distance \(d\). Given that \(\tan \alpha = 3\) and in terms of \(H\), the two possible values of \(d\). [6]

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]

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]

[In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in a vertical plane, \(\mathbf{i}\) being horizontal and \(\mathbf{j}\) being vertically upwards.] \includegraphics{figure_3} The point \(O\) is a fixed point on a horizontal plane. A ball is projected from \(O\) with velocity \((3\mathbf{i} + v\mathbf{j})\) m s\(^{-1}\) where \(v > 3\). The ball moves freely under gravity and passes through the point \(A\) before reaching its maximum height above the horizontal plane, as shown in Figure 3. The ball passes through \(A\) at time \(\frac{15}{49}\) s after projection. The initial kinetic energy of the ball is \(E\) joules. When the ball is at \(A\) it has kinetic energy \(\frac{1}{2}E\) joules.

1. Find the value of \(v\). [8]

At another point \(B\) on the path of the ball the kinetic energy is also \(\frac{1}{2}E\) joules. The ball passes through \(B\) at time \(T\) seconds after projection.

1. Find the value of \(T\). [3]

[In this question, the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in a vertical plane, \(\mathbf{i}\) being horizontal and \(\mathbf{j}\) being vertically upwards.] At time \(t = 0\), a particle \(P\) is projected with velocity \((4\mathbf{i} + 9\mathbf{j})\) m s\(^{-1}\) from a fixed point \(O\) on horizontal ground. The particle moves freely under gravity. When \(P\) is at the point \(H\) on its path, \(P\) is at its greatest height above the ground.

1. Find the time taken by \(P\) to reach \(H\). [2]

At the point \(A\) on its path, the position vector of \(P\) relative to \(O\) is \((k\mathbf{i} + k\mathbf{j})\) m, where \(k\) is a positive constant.

1. Find the value of \(k\). [4]
2. Find, in terms of \(k\), the position vector of the other point on the path of \(P\) which is at the same vertical height above the ground as the point \(A\). [3]

At time \(T\) seconds the particle is at the point \(B\) and is moving perpendicular to \((4\mathbf{i} + 9\mathbf{j})\)

1. Find the value of \(T\). [4]

The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) lie in a vertical plane, \(\mathbf{i}\) being horizontal and \(\mathbf{j}\) vertical. A ball of mass \(0.1\) kg is hit by a bat which gives it an impulse of \((3.5\mathbf{i} + 3\mathbf{j})\) Ns. The velocity of the ball immediately after being hit is \((10\mathbf{i} + 25\mathbf{j})\) m s\(^{-1}\).

1. Find the velocity of the ball immediately before it is hit. [3]

In the subsequent motion the ball is modelled as a particle moving freely under gravity. When it is hit the ball is 1 m above horizontal ground.

1. Find the greatest height of the ball above the ground in the subsequent motion. [3]

The ball is caught when it is again 1 m above the ground.

1. Find the distance from the point where the ball is hit to the point where it is caught. [4]

\includegraphics{figure_2} At time \(t = 0\) a small package is projected from a point \(B\) which is \(2.4\) m above a point \(A\) on horizontal ground. The package is projected with speed \(23.75\) m s\(^{-1}\) at an angle \(α\) to the horizontal, where \(\tan α = \frac{4}{5}\). The package strikes the ground at point \(C\), as shown in Fig. 2. The package is modelled as a particle moving freely under gravity.

1. Find the time taken for the package to reach \(C\). [5]

A lorry moves along the line \(AC\), approaching \(A\) with constant speed 18 m s\(^{-1}\). At time \(t = 0\) the rear of the lorry passes \(A\) and the lorry starts to slow down. It comes to rest 7 seconds later. The acceleration, \(a\) m s\(^{-2}\) of the lorry at time \(t\) seconds is given by $$a = -\frac{1}{4}t^2, \quad 0 \leq t \leq 7.$$

1. Find the speed of the lorry at time \(t\) seconds. [3]

1. Hence show that \(T = 6\). [3]

1. Show that when the package reaches \(C\) it is just under 10 m behind the rear of the moving lorry. [5]

END

\includegraphics{figure_3} A rocket \(R\) of mass 100 kg is projected from a point \(A\) with speed 80 m s\(^{-1}\) at an angle of elevation of \(60°\), as shown in Fig. 3. The point \(A\) is 20 m vertically above a point \(O\) which is on horizontal ground. The rocket \(R\) moves freely under gravity. At \(B\) the velocity of \(R\) is horizontal. By modelling \(R\) as a particle, find

1. the height in m of \(B\) above the ground, [4]

1. the time taken for \(R\) to reach \(B\) from \(A\). [2]

When \(R\) is at \(B\), there is an internal explosion and \(R\) breaks into two parts \(P\) and \(Q\) of masses 60 kg and 40 kg respectively. Immediately after the explosion the velocity of \(P\) is 80 m s\(^{-1}\) horizontally away from \(A\). After the explosion the paths of \(P\) and \(Q\) remain in the plane \(OAB\). Part \(Q\) strikes the ground at \(C\). By modelling \(P\) and \(Q\) as particles,

1. show that the speed of \(Q\) immediately after the explosion is 20 m s\(^{-1}\), [3]

1. find the distance \(OC\). [6]

END

A particle is projected from a point with speed \(u\) at an angle of elevation \(α\) above the horizontal and moves freely under gravity. When it has moved a horizontal distance \(x\), its height above the point of projection is \(y\).

1. Show that $$y = x \tan α - \frac{gx^2}{2u^2}(1 + \tan^2 α).$$ [5]

A shot-putter puts a shot from a point \(A\) at a height of 2 m above horizontal ground. The shot is projected at an angle of elevation of \(45°\) with a speed of 14 m s\(^{-1}\). By modelling the shot as a particle moving freely under gravity,

1. find, to 3 significant figures, the horizontal distance of the shot from \(A\) when the shot hits the ground, [5]

1. find, to 2 significant figures, the time taken by the shot in moving from \(A\) to reach the ground. [2]

A small smooth ball \(A\) of mass \(m\) is moving on a horizontal table with speed \(v\) when it collides directly with another small smooth ball \(B\) of mass \(3m\) which is at rest on the table. The balls have the same radius and the coefficient of restitution between the balls is \(e\). The direction of motion of \(A\) is reversed as a result of the collision.

1. Find, in terms of \(e\) and \(u\), the speeds of \(A\) and \(B\) immediately after the collision. [7]

In the subsequent motion \(B\) strikes a vertical wall, which is perpendicular to the direction of motion of \(B\), and rebounds. The coefficient of restitution between \(B\) and the wall is \(\frac{1}{2}\). Given that there is a second collision between \(A\) and \(B\),

1. find the range of values of \(e\) for which the motion described is possible. [6]

\includegraphics{figure_3} A rocket \(R\) of mass 100 kg is projected from a point \(A\) with speed 80 m s\(^{-1}\) at an angle of elevation of 60°, as shown in Fig. 3. The point \(A\) is 20 m vertically above a point \(O\) which is on horizontal ground. The rocket \(R\) moves freely under gravity. At \(B\) the velocity of \(R\) is horizontal. By modelling \(R\) as a particle, find

1. the height in m of \(B\) above the ground, [4]
2. the time taken for \(R\) to reach \(B\) from \(A\). [2]

When \(R\) is at \(B\), there is an internal explosion and \(R\) breaks into two parts \(P\) and \(Q\) of masses 60 kg and 40 kg respectively. Immediately after the explosion the velocity of \(P\) is 80 m s\(^{-1}\) horizontally away from \(A\). After the explosion the paths of \(P\) and \(Q\) remain in the plane \(OAB\). Part \(Q\) strikes the ground at \(C\). By modelling \(P\) and \(Q\) as particles,

1. show that the speed of \(Q\) immediately after the explosion is 20 m s\(^{-1}\), [3]
2. find the distance \(OC\). [6]

\includegraphics{figure_3} A ball \(B\) of mass 0.4 kg is struck by a bat at a point \(O\) which is 1.2 m above horizontal ground. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are respectively horizontal and vertical. Immediately before being struck, \(B\) has velocity \((-20\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\). Immediately after being struck it has velocity \((15\mathbf{i} + 16\mathbf{j})\) m s\(^{-1}\). After \(B\) has been struck, it moves freely under gravity and strikes the ground at the point \(A\), as shown in Fig. 3. The ball is modelled as a particle.

1. Calculate the magnitude of the impulse exerted by the bat on \(B\). [4]
2. By using the principle of conservation of energy, or otherwise, find the speed of \(B\) when it reaches \(A\). [6]
3. Calculate the angle which the velocity of \(B\) makes with the ground when \(B\) reaches \(A\). [4]
4. State two additional physical factors which could be taken into account in a refinement of the model of the situation which would make it more realistic. [2]

\includegraphics{figure_3} The object of a game is to throw a ball \(B\) from a point \(A\) to hit a target \(T\) which is placed at the top of a vertical pole, as shown in Figure 3. The point \(A\) is 1 m above horizontal ground and the height of the pole is 2 m. The pole is at a horizontal distance of 10 m from \(A\). The ball \(B\) is projected from \(A\) with a speed of 11 m s\(^{-1}\) at an angle of elevation of \(30°\). The ball hits the pole at the point \(C\). The ball \(B\) and the target \(T\) are modelled as particles.

1. Calculate, to 2 decimal places, the time taken for \(B\) to move from \(A\) to \(C\). [3]
2. Show that \(C\) is approximately 0.63 m below \(T\). [4]

The ball is thrown again from \(A\). The speed of projection of \(B\) is increased to \(V\) m s\(^{-1}\), the angle of elevation remaining \(30°\). This time \(B\) hits \(T\).

1. Calculate the value of \(V\). [6]
2. Explain why, in practice, a range of values of \(V\) would result in \(B\) hitting the target. [1]

\includegraphics{figure_3} A particle \(P\) is projected from a point \(A\) with speed \(u\) m s\(^{-1}\) at an angle of elevation \(\theta\), where \(\cos \theta = \frac{4}{5}\). The point \(B\), on horizontal ground, is vertically below \(A\) and \(AB = 45\) m. After projection, \(P\) moves freely under gravity passing through a point \(C\), 30 m above the ground, before striking the ground at the point \(D\), as shown in Figure 3. Given that \(P\) passes through \(C\) with speed 24.5 m s\(^{-1}\),

1. using conservation of energy, or otherwise, show that \(u = 17.5\), [4]
2. find the size of the angle which the velocity of \(P\) makes with the horizontal as \(P\) passes through \(C\), [3]
3. find the distance \(BD\). [7]