Exercise 3 Motion in two dimensions [12 points]

A apple is hanging from a height h and an archer on the ground d meters away from the tree wants to hit it with a arrow. He aims straight at the apple and releases the arrow. Exactly at the same time the apple starts falling from the tree. The initial speed of the arrow is v.

a) At what time does the trajectory of the arrow cross the trajectory of the apple? What height do apple and arrow have at that time? Make a sketch of the trajectories.

b) Show that the archer will hit the apple provided that the initial speed of the arrow obeys v ≥ (d2 +h2 )g. What does this requirement ensure?

2h

c) Assuming the archer does not aim at the apple, what is the optimal angle to shoot the arrow so that it covers the maximal horizontal distance? What is the range of the arrow in this case? Hint: You might need the trigonometric identity 2 cos φ sin φ = sin 2φ.

Note that we neglect friction, the curvature of the Earth, and the finite size of arrow and apple, which are both assumed to be material points in this problem