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A particle P is projected from a point A with speed u at an angle of elevation  and moves freely under gravity.
a. Show that the greatest height of P above the level of A is ( u 2 sin 2  ) / 2g.
In using a sophisticated rocket launcher, both the angle of elevation and the initial speed of the rockets can be adjusted. In a battle, one army wishes to use its launcher to hit an enemy arms factory. The launcher is positioned at a point which is 6 km away from the factory and at the same horizontal level. In order to avoid the enemy's anti-missile detection devices, the rocket must be launched in such a way that its maximum height during flight is 200 m above the ground, which may be assumed to be horizontal. The rocket is modelled as a particle moving freely under gravity, and the arms factory and the launcher as points on the level of the ground. The rocket is launched at an angle  to the horizontal in such a way as to hit the factory.
b. Show that tan  = 2/15.
c. Find the initial speed of the rocket, giving your answer in ms-1 to the nearest whole number.
d. Find the time taken for the rocket to reach the factory, giving your answer in seconds to 1 decimal place.

3. A particle P is projected from a point A on horizontal ground, with speed u at an angle of elevation , and moves freely under gravity.
c. Show that P hits the ground at a point B where AB = ( 2u 2 sin  cos &#61537 / g
An athlete throws a javelin which lands at a horizontal distance of 61.25 m from where it is thrown. The initial angle of projection of the javelin is 45° above the horizontal. The javelin is initially modelled as a particle projected from ground level and moving freely under gravity. Using this model,
d. find the initial speed of the javelin,
e. find, in s to 1 decimal place, the time of flight of the javelin.
A refinement of the model takes into account the fact that the javelin is projected from a point above the ground, rather than from ground level. The initial angle of projection and the horizontal distance travelled are unaltered.
d. State, with a brief reason, whether your answer to part (b) would be increased or decreased as a result of using this refinement of the model.
[11]
2. A particle P moves along the x -axis. It passes through the origin O at time t = 0 with speed 15 ms-1 in the direction of x increasing. At time t seconds the acceleration of P in the direction of x increasing is (6t – 18) ms-2.
a. Find the values of t at which P is instantaneously at rest.
b. Find the distance between the points at which P is instantaneously at rest.
[12 marks]
3. Two motor boats A and B are moving with constant velocities. The velocity of A is 30 kmh-1 due north, and B is moving at 20 kmh-1 on a bearing of 060o. The unit vectors i and j are directed due east and north respectively. At 10 am, the position vector of B is 70j km relative to a fixed origin O and A is at the point O ; t hours later, the position vectors of A and B, are r km and s km respectively.
a. Find the velocity of B in the form (p i + q j) kmh-1.
b. Find expressions for r and s in terms of t.
The boats can maintain radio contact with each other, provided that the distance between them is no more than 70 km.
c. Find the time at which the boats are again at the maximum distance at which they can maintain radio contact with each other.
[16 marks]
4. A particle P moves along the Ox axis. Its velocity, v ms-1, t seconds after leaving the origin O , is given by v = 12 + 4t – t 2 Find
a. the acceleration of P when its velocity is zero,
b. the distance of P from O when its acceleration is zero,
c. the total distance travelled by P in the interval 0  t  9.
[18 marks]
Attached Files
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