Time when specific condition met

1 A small ball \(B\) is projected with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. At the instant 0.8 s after projection, \(B\) is 0.5 m vertically above the top of a vertical post.

1. Calculate the height of the top of the post above the ground.
2. Show that \(B\) is at its greatest height 0.2 s before passing over the post.

3 A stone is thrown with speed \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal from a point on horizontal ground. Find the distance between the two points at which the path of the stone makes an angle of \(45 ^ { \circ }\) with the horizontal.

4 A stone is projected from a point on horizontal ground with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal, where \(\sin \theta = \frac { 4 } { 5 }\). At time 1.2 s after projection the stone passes through the point \(A\). Subsequently the stone passes through the point \(B\), which is at the same height above the ground as \(A\). Find the horizontal distance \(A B\).

1 A particle is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal. Calculate the time after projection when the particle has speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is rising.

1 A particle is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal. Calculate the time after projection when the particle has speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is rising.

7 A missile is projected from a point \(O\) on horizontal ground with speed \(175 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\theta\). The horizontal lower surface of a cloud is 650 m above the ground.

1. Find the value of \(\theta\) for which the missile just reaches the cloud. It is given that \(\theta = 55 ^ { \circ }\).
2. Find the length of time for which the missile is above the lower surface of the cloud.
3. Find the speed of the missile at the instant it enters the cloud.

8 A particle is launched from the point \(A\) on a horizontal surface, with a velocity of \(22.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{828e8db1-efcf-4878-8292-ba5bbd80115c-5_369_1182_406_431} After 2 seconds, the particle reaches the point \(C\), where it is at its maximum height above the surface.

1. Show that \(\sin \theta = 0.875\).
2. Find the height of the point \(C\) above the horizontal surface.
3. The particle returns to the surface at the point \(B\). Find the distance between \(A\) and \(B\). (3 marks)
4. Find the length of time during which the height of the particle above the surface is greater than 5 metres.
5. Find the minimum speed of the particle.

7 At time \(t = 0\), a small ball \(A\) is projected vertically upwards with speed \(8 \mathrm {~ms} ^ { - 1 }\) from a fixed point on horizontal ground.
The ball hits the ground again for the first time at time \(t = T _ { 1 }\) seconds.
Ball \(A\) is modelled as a particle moving freely under gravity.

1. Show that \(T _ { 1 } = 1.63\) to 3 significant figures. After the first impact with the ground, \(A\) rebounds to a height of 2 m above the ground.
   Given that the mass of \(A\) is 0.1 kg ,
2. find the magnitude of the impulse received by \(A\) as a result of its first impact with the ground. At time \(t = 1\) second, another small ball \(B\) is projected vertically upwards from another point on the ground with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Ball \(B\) is modelled as a particle moving freely under gravity.
   At time \(t = T _ { 2 }\) seconds ( \(T _ { 2 } > 1\) ), \(A\) and \(B\) are at the same height above the ground for the first time.
3. Find the value of \(T _ { 2 }\)

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 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 small ball is projected with speed \(16 \text{ ms}^{-1}\) at an angle of \(45°\) above the horizontal from a point on horizontal ground. Calculate the period of time, before the ball lands, for which the speed of the ball is less than \(12 \text{ ms}^{-1}\). [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]

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]