How high will an arrow travel if it is fired straight up, and what is the Potential Energy it would have at the peak?

Question

The bows mass is 0.37 kg

The arrows mass is 0.039 kg

The gravity is 9.8 m/s^2

The stretch (x) is 50 cm (0.5 meters)

The Fg = Fspring = 38N

Answer 1

Hi Spencer O.,

An interesting problem. Not for the mechanics of solution, but for the assumptions you must choose to make first.

So -- you can calculate the potential energy stored in the stretched bow: (1/2) k x^2, however you name your variables! But, when the arrow is fired, is that PE completely transferred to the arrow, or should the bow + arrow be treated as a "floating" unit? Thus, the bow might move downwards as the arrow moves upwards, for a net momentum change of zero. That would imply a sightly lower initial speed of the arrow vs. the ground? This "red-herring" possibility is dangled by providing a mass for the bow!

However, I think that usually (i.e. if a hunter, rather than a physicist, is firing the arrow) the bow is fixed in position by the archer; and the arrow does carry the entire PE with it as KE, initially.

So all you need to do is equate gravitational PE to initial KE, and solve.

P.S. If you are an observer at this event, you might want to clear out of the way. When the arrow comes down, you are likely to get skewered. Just sayin'.

-- Cheers, -- Mr. d.