Bow and Arrow

Use conservation of energy to predict the height the arrow will reach.

bow and arrow

What it shows:

When the string of a bow and arrow is pulled from equilibrium, the elastic potential energy in the bow is converted to kinetic energy of the arrow when the string is released. When the arrow reaches the top of its flight, it has zero kinetic energy and the initial elastic potential energy is now gravitational potential energy of the arrow. Using conservation of energy, one can calculate how far the bow string must be pulled back in order for the arrow to reach a certain height (specifically, for the arrow to just touch the ceiling of the lecture hall).

How it works:

For safety purposes, the arrow has a tennis ball glued to its tip. With the bow pointing up and secured to a cart as pictured, weights are hung from the bow string. Measuring the displacement of the bow string as more weights are added shows that the bow obeys Hooke's Law. The graph shows displacement as a function of force. It is a linear relationship for reasonable displacements and the bow's stiffness, or force constant, can be deduced from the slope of the line. Knowing the force constant, one can then calculate how much work is done when the bow string is pulled a certain displacement. Alternatively, it's equal to the area under the curve.

Setting it up:

Use the "bow stringer" to string the bow (a 35# recurve York Crest bow of 64" length) with the 60" bowstring. The bow then gets clamped into a special mount made for this experiment. The mount may be attached to one of our demo carts or to the lecture bench. Once its location has been decided upon, one needs to measure (using the HiJacker) the distance from the tip of the arrow to the ceiling.

Having performed all the necessary measurements to determine the arrow's mass, the force constant of the bow, and the height one would like to shoot the arrow (all of which can be done before class if time is an issue), attach the trigger-release to the bow string and pull it back the calculated distance. The arrow is released when the trigger is touched.


Plotting the displacement of the bow string as successive 1 kg weights are added shows the force constant of the bow to be about 500 N/m.


As an example, pulling the bow string back 12.23 cm should propel the 87.57-gram arrow about 4.217 m high to just barely touch the ceiling. And it does!