Two particles: different start times, same height

5 A particle is projected vertically upwards from a point \(O\) with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Two seconds later a second particle is projected vertically upwards from \(O\) with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after the second particle is projected, the two particles collide.

1. Find \(t\). \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-06_65_1569_488_328}
2. Hence find the height above \(O\) at which the particles collide.

4 A ball \(A\) is released from rest at the top of a tall tower. One second later, another ball \(B\) is projected vertically upwards from ground level near the bottom of the tower with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The two balls are at the same height 1.5 s after ball \(B\) is projected.

1. Show that the height of the tower is 50 m .
2. Find the length of time for which ball \(B\) has been in motion when ball \(A\) reaches the ground. Hence find the total distance travelled by ball \(B\) up to the instant when ball \(A\) reaches the ground.

3. A particle, \(P\), is projected vertically upwards with speed \(U\) from a fixed point \(O\). At the instant when \(P\) reaches its greatest height \(H\) above \(O\), a second particle, \(Q\), is projected with speed \(\frac { 1 } { 2 } U\) vertically upwards from \(O\).

1. Find \(H\) in terms of \(U\) and \(g\).
2. Find, in terms of \(U\) and \(g\), the time between the instant when \(Q\) is projected and the instant when the two particles collide.
3. Find where the two particles collide. DO NOT WRITEIN THIS AREA \includegraphics[max width=\textwidth, alt={}, center]{916543cb-14f7-486c-ba3c-eda9be134045-08_2666_99_107_1957}

1. A small stone is released from rest from a point \(A\) which is at height \(h\) metres above horizontal ground. Exactly one second later another small stone is projected with speed \(19.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards from a point \(B\), which is also at height \(h\) metres above the horizontal ground. The motion of each stone is modelled as that of a particle moving freely under gravity. The two stones hit the ground at the same time.

Find the value of \(h\).

1. At time \(t = 0\), a stone is thrown vertically upwards with speed \(19.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) which is \(h\) metres above horizontal ground. At time \(t = 3 \mathrm {~s}\), another stone is released from rest from a point \(B\) which is also \(h\) metres above the same horizontal ground. Both stones hit the ground at time \(t = T\) seconds. The motion of each stone is modelled as that of a particle moving freely under gravity.

Find

1. the value of \(T\),
2. the value of \(h\).
   

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4 A particle is projected vertically upwards from a point O at \(21 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the greatest height reached by the particle. When this particle is at its highest point, a second particle is projected vertically upwards from \(O\) at \(15 \mathrm {~ms} ^ { - 1 }\).
2. Show that the particles collide 1.5 seconds later and determine the height above O at which the collision takes place.

2 A particle is projected vertically upwards from a point O at \(21 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the greatest height reached by the particle. When this particle is at its highest point, a second particle is projected vertically upwards from \(O\) at \(15 \mathrm {~ms} ^ { - 1 }\).
2. Show that the particles collide 1.5 seconds later and determine the height above O at which the collision takes place.