3.02i Projectile motion: constant acceleration model

A particle \(P\) is projected with speed \(20 \text{ m s}^{-1}\) at an angle of \(60°\) below the horizontal, from a point \(O\) which is \(30 \text{ m}\) above horizontal ground.

1. Calculate the time taken by \(P\) to reach the ground. [3]
2. Calculate the speed and direction of motion of \(P\) immediately before it reaches the ground. [4]

\includegraphics{figure_7} A particle \(P\) is projected from a point \(O\) with initial speed \(10 \text{ m s}^{-1}\) at an angle of \(45°\) above the horizontal. \(P\) subsequently passes through the point \(A\) which is at an angle of elevation of \(30°\) from \(O\) (see diagram). At time \(t \text{ s}\) after projection the horizontal and vertically upward displacements of \(P\) from \(O\) are \(x \text{ m}\) and \(y \text{ m}\) respectively.

1. Write down expressions for \(x\) and \(y\) in terms of \(t\), and hence obtain the equation of the trajectory of \(P\). [3]
2. Calculate the value of \(x\) when \(P\) is at \(A\). [3]
3. Find the angle the trajectory makes with the horizontal when \(P\) is at \(A\). [4]

A particle \(P\) is projected with speed \(26\) m s\(^{-1}\) at an angle of \(30°\) above the horizontal from a point \(O\) on a horizontal plane.

1. For the instant when the vertical component of the velocity of \(P\) is \(5\) m s\(^{-1}\) downwards, find the direction of motion of \(P\) and the height of \(P\) above the plane. [4]
2. \(P\) strikes the plane at the point \(A\). Calculate the time taken by \(P\) to travel from \(O\) to \(A\) and the distance \(OA\). [3]

\includegraphics{figure_2} A particle \(P\) is projected from a point \(O\) at an angle of \(60°\) above horizontal ground. At an instant 0.6 s after projection, the angle of elevation of \(P\) from \(O\) is \(45°\) (see diagram).

1. Show that the speed of projection of \(P\) is 8.20 m s\(^{-1}\), correct to 3 significant figures. [4]
2. Calculate the time after projection when the direction of motion of \(P\) is \(45°\) above the horizontal. [3]

A particle \(P\) is projected with speed \(30\) m s\(^{-1}\) at an angle of \(60°\) above the horizontal from a point \(O\) on horizontal ground. For the instant when the speed of \(P\) is \(17\) m s\(^{-1}\) and increasing,

1. show that the vertical component of the velocity of \(P\) is \(8\) m s\(^{-1}\) downwards, [2]
2. calculate the distance of \(P\) from \(O\). [5]

A particle \(P\) is projected with speed \(V\) m s\(^{-1}\) at an angle of \(30°\) above the horizontal from a point \(O\) on horizontal ground. At the instant \(2\) s after projection, \(OP\) makes an angle of \(15°\) above the horizontal. Calculate \(V\). [4]

The equation of the trajectory of a small ball \(B\) projected from a fixed point \(O\) is $$y = -0.05x^2,$$ where \(x\) and \(y\) are, respectively, the displacements in metres of \(B\) from \(O\) in the horizontal and vertically upwards directions.

1. Show that \(B\) is projected horizontally, and find its speed of projection. [3]
2. Find the value of \(y\) when the direction of motion of \(B\) is \(60°\) below the horizontal, and find the corresponding speed of \(B\). [6]

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]

A particle \(P\) is projected with speed 35 m s\(^{-1}\) from a point \(O\) on a horizontal plane. In the subsequent motion, the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x\) m and \(y\) m respectively. The equation of the trajectory of \(P\) is $$y = kx - \frac{(1 + k^2)x^2}{245},$$ where \(k\) is a constant. \(P\) passes through the points \(A(14, a)\) and \(B(42, 2a)\), where \(a\) is a constant.

1. Calculate the two possible values of \(k\) and hence show that the larger of the two possible angles of projection is 63.435°, correct to 3 decimal places. [5]

For the larger angle of projection, calculate

1. the time after projection when \(P\) passes through \(A\), [2]
2. the speed and direction of motion of \(P\) when it passes through \(B\). [4]

A small ball \(B\) is projected with speed \(30\text{ m s}^{-1}\) at an angle of \(60°\) to the horizontal from a point on horizontal ground. Find the time after projection when the speed of \(B\) is \(25\text{ m s}^{-1}\) for the second time. [4]

\includegraphics{figure_7} A small object is projected with speed \(24\text{ m s}^{-1}\) from a point \(O\) at the foot of a plane inclined at \(45°\) to the horizontal. The angle of projection of the object is \(15°\) above a line of greatest slope of the plane (see diagram). At time \(t\) s after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x\) m and \(y\) m respectively.

1. Express \(x\) and \(y\) in terms of \(t\), and hence find the value of \(t\) for the instant when the object strikes the plane. [4]
2. Express the vertical height of the object above the plane in terms of \(t\) and hence find the greatest vertical height of the object above the plane. [5]

\includegraphics{figure_7} A small object is projected with speed \(24\text{ m s}^{-1}\) from a point \(O\) at the foot of a plane inclined at \(45°\) to the horizontal. The angle of projection of the object is \(15°\) above a line of greatest slope of the plane (see diagram). At time \(t\text{ s}\) after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x\text{ m}\) and \(y\text{ m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\), and hence find the value of \(t\) for the instant when the object strikes the plane. [4]
2. Express the vertical height of the object above the plane in terms of \(t\) and hence find the greatest vertical height of the object above the plane. [5]

A particle \(P\) is projected with speed \(u\) at an angle of \(30°\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The particle reaches its greatest height at time \(T\) after projection. Find, in terms of \(u\), the speed of \(P\) at time \(\frac{2}{3}T\) after projection. [5]

A particle \(P\) is projected with speed \(u\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The direction of motion of \(P\) makes an angle \(\alpha\) above the horizontal when \(P\) first reaches three-quarters of its greatest height.

1. Show that \(\tan \alpha = \frac{1}{2}\tan \theta\). [6]

A particle \(P\) is projected from a point \(O\) on a horizontal plane and moves freely under gravity. The initial velocity of \(P\) is 100 ms\(^{-1}\) at an angle \(\theta\) above the horizontal, where \(\tan\theta = \frac{4}{3}\). The two times at which \(P\)'s height above the plane is \(H\) m differ by 10 s.

1. Find the value of \(H\). [5]

1. Find the magnitude and direction of the velocity of \(P\) one second before it strikes the plane. [4]

A particle \(P\) is projected with speed \(u\) at an angle \(\theta\) 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 \(t\) are denoted by \(x\) and \(y\) respectively.

1. Use the equation of the trajectory given in the List of formulae (MF19), together with the condition \(y = 0\), to establish an expression for the range \(R\) in terms of \(u\), \(\theta\) and \(g\). [2]
2. Deduce an expression for the maximum height \(H\), in terms of \(u\), \(\theta\) and \(g\). [2]

It is given that \(R = \frac{4H}{\sqrt{3}}\).

1. Show that \(\theta = 60°\). [1]

It is given also that \(u = \sqrt{40}\) m s\(^{-1}\).

1. Find, by differentiating the equation of the trajectory or otherwise, the set of values of \(x\) for which the direction of motion makes an angle of less than \(45°\) with the horizontal. [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]

At time \(t\)s, a particle \(P\) is projected with speed \(40\)m s\(^{-1}\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The greatest height achieved by \(P\) during its flight is \(H\)m and the corresponding time is \(T\)s.

1. Obtain expressions for \(H\) and \(T\) in terms of \(\theta\). [2]

During the time between \(t = T\) and \(t = 3\), \(P\) descends a distance \(\frac{1}{4}H\).

1. Find the value of \(\theta\). [4]
2. Find the speed of \(P\) when \(t = 3\). [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]

At time \(t = 0\) seconds, a particle \(P\) is projected with speed \(u\) m s\(^{-1}\) at an angle \(60°\) above the horizontal from a point \(O\). In the subsequent motion \(P\) moves freely under gravity. The direction of motion of \(P\) when \(t = 5\) is perpendicular to its direction of motion when \(t = 15\). Find the value of \(u\). [5]

A smooth sphere with centre \(O\) and of radius \(a\) is fixed to a horizontal plane. A particle \(P\) of mass \(m\) is projected horizontally from the highest point of the sphere with speed \(u\), so that it begins to move along the surface of the sphere. The particle \(P\) loses contact with the sphere at the point \(Q\) on the sphere, where \(OQ\) makes an angle \(\theta\) with the upward vertical through \(O\).

1. Show that \(\cos\theta = \frac{u^2 + 2ag}{3ag}\). [4]

It is given that \(\cos\theta = \frac{5}{9}\).

1. Find, in terms of \(a\) and \(g\), an expression for the vertical component of the velocity of \(P\) just before it hits the horizontal plane to which the sphere is fixed. [3]
2. Find an expression for the time taken by \(P\) to fall from \(Q\) to the plane. Give your answer in the form \(k\sqrt{\frac{a}{g}}\), stating the value of \(k\) correct to 3 significant figures. [2]

A particle \(P\) is projected with speed \(u\text{ ms}^{-1}\) at an angle \(\theta\) above the horizontal from a point \(O\) and moves freely under gravity. After 5 seconds the speed of \(P\) is \(\frac{3}{4}u\).

1. Show that \(\frac{7}{16}u^2 - 100u\sin\theta + 2500 = 0\). [3]
2. It is given that the velocity of \(P\) after 5 seconds is perpendicular to the initial velocity. Find, in either order, the value of \(u\) and the value of \(\sin\theta\). [5]