A particle \(Q\) is initially positioned at a distance \(d\) vertically above a particle \(P\). Particle \(P\) is projected with speed \(U\) at an angle \(\alpha\) above the horizontal. At the same time, \(Q\) is projected at an angle \(\beta\) below the horizontal. Both particles move freely under gravity. The particles collide at time \(T\) after the projections. (a) Show that \(d=U T(\sin \alpha+\cos \alpha \tan \beta)\).

The particles collide when \(P\) is at its maximum height. (b) Given that \(\alpha=30^{\circ}\) and \(\beta=60^{\circ}\), find \(d\) in terms of \(U\) and \(g\).

A particle \(P\) is projected from \(O\) with speed \(U\) at angle \(45^\circ\) above the horizontal and moves freely under gravity.

(a) State the vertical and horizontal components of velocity at time \(t\).

At time \(T\), \(P\) is moving at angle \(60^\circ\) below the horizontal.

(b) Show that \(T=\dfrac{U}{2g}(\sqrt2+\sqrt6)\).

At time \(T\), the particle strikes a smooth horizontal plane at a point a horizontal distance \(D\) from \(O\) and a vertical distance \(H\) below \(O\).

(c) Find \(H:D\).

After striking the plane, \(P\) rebounds with speed \(w\). The coefficient of restitution is \(\frac23\).

(d) Find \(w\) in terms of \(U\).

At time \(t=0\), a particle \(P\) is projected with speed \(u\,\text{m s}^{-1}\) at an angle \(60^\circ\) above the horizontal from a point \(O\). It then moves freely under gravity.

The direction of motion of \(P\) when \(t=5\) is perpendicular to its direction of motion when \(t=15\).

Find \(u\).

A particle \(P\) is projected with speed \(u\,\text{m s}^{-1}\) at an angle \(\theta\) above the horizontal from a point \(O\), and moves freely under gravity.

After \(5\) seconds the speed of \(P\) is \(\dfrac34u\).

(a) Show that

(b) It is given that the velocity of \(P\) after \(5\) seconds is perpendicular to the initial velocity. Find, in either order, \(u\) and \(\sin\theta\).

At time \(t \mathrm{~s}\), a particle \(P\) is projected with speed \(40 \mathrm{~ms}^{-1}\) 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 \(H \mathrm{~m}\) and the corresponding time is \(T \mathrm{~s}\).
(a) Obtain expressions for \(H\) and \(T\) in terms of \(\theta\).

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

(c) Find the speed of \(P\) when \(t=3\).

The points \(O\) and \(P\) are on a horizontal plane, a distance 8 m apart. A ball is thrown from \(O\) with speed \(u \mathrm{~ms}^{-1}\) at an angle \(\theta\) above the horizontal, where \(\tan \theta=\frac{4}{3}\). At the same instant, a model aircraft is launched with speed \(5 \mathrm{~ms}^{-1}\) 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 \(5 \mathrm{~m} \mathrm{~s}^{-1}\). After \(T \mathrm{~s}\), the ball and the model aircraft collide.
(a) Find the value of \(T\).
(b) Find the direction in which the ball is moving immediately before the collision.

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 \(\frac{35}{2} \mathrm{~ms}^{-1}\) at an angle \(\alpha\) above the horizontal, where \(\tan \alpha=\frac{4}{3}\). Particle \(Q\) is projected with speed \(u \mathrm{~ms}^{-1}\) 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 \(T \mathrm{~s}\) after the projection of particle \(Q\).
(a) Write down expressions, in terms of \(T\), for the horizontal displacements of \(P\) and \(Q\) from \(O\) when they collide and hence show that \(4 u T=21 \sqrt{5}(T+1)\).
(b) Find the value of \(T\).
(c) Find the horizontal and vertical displacements of the particles from \(O\) when they collide.

A particle \(P\) is projected with speed \(25 \mathrm{~ms}^{-1}\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. After 2 s the speed of \(P\) is \(15 \mathrm{~ms}^{-1}\).
(a) Find the value of \(\sin \theta\).

(b) Find the range of the flight.

A particle \(P\) is projected with speed \(V\,\mathrm{m\,s}^{-1}\) at an angle \(75^{\circ}\) above the horizontal from a point \(O\) on a horizontal plane. It then moves freely under gravity.
(a) Show that the total time of flight, in seconds, is \(\frac{2 V}{g} \sin 75^{\circ}\).

A smooth vertical barrier is now inserted with its lower end on the plane at a distance 15 m from \(O\). The particle is projected as before but now strikes the barrier, rebounds and returns to \(O\). The coefficient of restitution between the barrier and the particle is \(\frac{3}{5}\).
(b) Explain why the total time of flight is unchanged.
(c) Find an expression for \(V\) in terms of \(g\).

A particle \(P\) is projected from a point \(O\) on a horizontal plane and moves freely under gravity. The initial velocity of \(P\) is \(100 \mathrm{~ms}^{-1}\) 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 \(H \mathrm{~m}\) differ by 10 s .
(a) Find the value of \(H\).
(b) Find the magnitude and direction of the velocity of \(P\) one second before it strikes the plane.

A particle \(P\) is projected from a point \(O\) on a horizontal plane and moves freely under gravity. Its initial speed is \(u \mathrm{~m} \mathrm{~s}^{-1}\) and its angle of projection is \(\sin ^{-1}\left(\frac{4}{5}\right)\) 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 \(u\).

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.
(a) Write down the horizontal and vertical components of the velocity of the particle at time \(T\) after projection.

At time \(T\) after projection, the direction of motion of the particle is perpendicular to the direction of projection.
(b) Express \(T\) in terms of \(u, g\) and \(\alpha\).
(c) Deduce that \(T\gt \frac{u}{g}\).

A particle \(P\) is projected with speed \(u\) at an angle of \(30^{\circ}\) 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.

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.
(a) Show that \(\tan \alpha=\frac{1}{2} \tan \theta\).
(b) Given that \(\tan \theta=\frac{4}{3}\), find the horizontal distance travelled by \(P\) when it first reaches three-quarters of its greatest height. Give your answer in terms of \(u\) and \(g\).