A ball is thrown from a position \(O\) on level ground with initial velocity (in \(\mathrm{m\,s^{-1}}\)) \(\mathbf{u}=5\mathbf{i}+6\mathbf{j}\) where \(\mathbf{i}\) is a horizontal direction and \(\mathbf{j}\) is vertically upwards.

If the wind is blowing horizontally in the direction shown, it will change the trajectory of the ball. Sketch what you think the trajectory might look like.

How will the range of the throw be affected?

What effect will the wind have on the maximum height of the ball?

Will the trajectory still be the same sort of shape?

Is it possible for the ball to be blown back past the thrower?

One way to model the effect of the wind is as a constant horizontal acceleration. Let’s call the magnitude of this acceleration \(\quantity{w}{m\,s^{-2}}\).

Find how far from \(O\) the ball will land as a function of \(w\). Express the trajectory of the ball using parametric equations for \(x\) and \(y\) in terms of time, \(t\).

Take gravity to be \(\quantity{10}{m\,s^{-2}}\).

In each of these cases, find where the ball lands and draw a sketch of the trajectory:

\(w=0\)

\(w=2.5\)

\(w=5\)

\(w=10\)

What stays the same and what changes as we vary the value of \(w\)?

What value of \(w\) would make the ball return to \(O\)?