- The surface of a horizontal tennis court is modelled as part of a horizontal plane, with the origin on the ground at the centre of the court, and
- i and j are unit vectors directed across the width and length of the court respectively
- \(\quad \mathbf { k }\) is a unit vector directed vertically upwards
- units are metres
After being hit, a tennis ball, modelled as a particle, moves along the path with equation
$$\mathbf { r } = \left( - 4.1 + 9 \lambda - 2.3 \lambda ^ { 2 } \right) \mathbf { i } + ( - 10.25 + 15 \lambda ) \mathbf { j } + \left( 0.84 + 0.8 \lambda - \lambda ^ { 2 } \right) \mathbf { k }$$
where \(\lambda\) is a scalar parameter with \(\lambda \geqslant 0\)
Assuming that the tennis ball continues on this path until it hits the ground,
- find the value of \(\lambda\) at the point where the ball hits the ground.
The direction in which the tennis ball is moving at a general point on its path is given by
$$( 9 - 4.6 \lambda ) \mathbf { i } + 15 \mathbf { j } + ( 0.8 - 2 \lambda ) \mathbf { k }$$
- Write down the direction in which the tennis ball is moving as it hits the ground.
- Hence find the acute angle at which the tennis ball hits the ground, giving your answer in degrees to one decimal place.
The net of the tennis court lies in the plane \(\mathbf { r } . \mathbf { j } = 0\)
- Find the position of the tennis ball at the point where it is in the same plane as the net.
The maximum height above the court of the top of the net is 0.9 m .
Modelling the top of the net as a horizontal straight line, - state whether the tennis ball will pass over the net according to the model, giving a reason for your answer.
With reference to the model,
- decide whether the tennis ball will actually pass over the net, giving a reason for your answer.