Vertical projection: max height

2 A particle \(P\) is projected vertically upwards from horizontal ground with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 } . P\) reaches a maximum height of 20 m above the ground.

1. Find the value of \(u\).
2. Find the total time for which \(P\) is at least 15 m above the ground.

1 A particle \(P\) is projected vertically upwards with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on the ground. \(P\) reaches its greatest height after 3 s .

1. Find \(v\).
2. Find the greatest height of \(P\) above the ground.

1 A particle \(P\) is projected vertically upwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on the ground. \(P\) reaches its greatest height after 3 s .

1. Find \(u\).
2. Find the greatest height of \(P\) above the ground.

1 A particle \(P\) is projected vertically upwards with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on the ground.

1. Find the greatest height above the ground reached by \(P\).
2. Find the total time from projection until \(P\) returns to the ground.

3 A particle \(P\) is projected vertically upwards from a point \(O\). When the particle is at a height of 0.5 m , its speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the greatest height reached by the particle above \(O\),
2. the time after projection at which the particle returns to \(O\).

1 A particle is thrown vertically upwards and returns to its point of projection after 6 seconds. Air resistance is negligible. Calculate the speed of projection of the particle and also the maximum height it reaches.

1. At time \(t = 0\), a small stone is thrown vertically upwards with speed \(14.7 \mathrm {~ms} ^ { - 1 }\) from a point \(A\).

At time \(t = T\) seconds, the stone passes through \(A\), moving downwards.
The stone is modelled as a particle moving freely under gravity throughout its motion.
Using the model,

1. find the value of \(T\),
2. find the total distance travelled by the stone in the first 4 seconds of its motion.
3. State one refinement that could be made to the model, apart from air resistance, that would make the model more realistic.

3 A particle is thrown vertically upwards and returns to its point of projection after 6 seconds. Air resistance is negligible. Calculate the speed of projection of the particle and also the maximum height it reaches.

8 A small ball is thrown vertically upwards with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), from a point 5 m above the ground. Assuming air resistance is negligible, find

1. the greatest height above the ground reached by the ball,
2. the time taken for the ball to reach the ground.

A particle \(P\) of mass \(0.4\) kg is projected vertically upwards from horizontal ground with speed \(10\) m s\(^{-1}\).

1. Find the greatest height above the ground reached by \(P\). [2]

When \(P\) reaches the ground again, it bounces vertically upwards. At the first instant that it hits the ground, \(P\) loses \(7.2\) J of energy.

1. Find the time between the first and second instants at which \(P\) hits the ground. [4]

A stone is projected vertically upwards from ground level at a speed of \(30\,\mathrm{m}\,\mathrm{s}^{-1}\). It is assumed that there is no wind or air resistance. Find the maximum height it reaches and the total time it takes from its projection to its return to ground level. [5]

A stone is projected vertically upwards from ground level at a speed of \(30 \mathrm{~m} \mathrm{~s}^{-1}\). It is assumed that there is no wind or air resistance. Find the maximum height it reaches and the total time it takes from its projection to its return to ground level. [5]

A stone is projected vertically upwards from ground level at a speed of \(30 \mathrm{m} \mathrm{s}^{-1}\). It is assumed that there is no wind or air resistance. Find the maximum height it reaches and the total time it takes from its projection to its return to ground level. [5]