Conceptualizing units. A small frozen burrito, like those bought at your local convenience...

Conceptualizing units. A small frozen burrito, like those bought at your local convenience store, has a mass of approximately 100 g, which gives it a weight of roughly 1 newton. (a) From what height would you need to drop the burrito to give it 1 j of kinetic energy when it hits the ground? How many burritos would you have to drop from that height to equal the kinetic energy of (b) the arrow in Example 6.7 and (c) the snowboarder (m = 65 kg) in Figure 6.12?

Figure 6.12 A and B These two hills have the same height h, so even though they have very different shapes, the work done by gravity in the two cases is the same: W[hill A] = W[hill B]. The change in the potential energy is therefore the same in the two cases (ΔPEa = ΔPEb). C These bar charts show the initial and final kinetic and potential energies of the snowboarder.

Example 6.7

Spring Constant of an Archer’s Bow

Consider a professional archer engaged in target practice. An archer’s bow behaves like a spring and is described by Hooke’s law. The force exerted by the bow on the arrow is proportional to the displacement of the bow string as shown in Figure 6.25. Suppose the archer is able to shoot an arrow of mass m = 0.050 kg with a velocity of 100 m/s just as it leaves the bow. What is the spring constant of her bow?

Figure 6.25 Example 6.7. A A bow acts like a spring, and its potential energy is given by , where x is the distance the bowstring is displaced. B Bar charts showing the initial and final kinetic and potential energies of the arrow and bow.

RECOGNIZE THE PRINCIPLE

We want to apply the conservation of energy condition to the combination of the bow plus arrow. The initial state has the bow fully stretched (the bow string pulled back a distance xi and the arrow at rest. The final state has the bow in its relaxed state (xf = 0) and the arrow leaving the bow with a velocity vf = 100 m/s.

SKETCH THE PROBLEM

The problem is shown in Figure 6.25A.

IDENTIFY THE RELATIONSHIPS

Writing the conservation of energy condition with the elastic potential energy of the bow, we have

SOLVE

Inserting our values for the initial arrow velocity (vi = 0) and final bow stretch (xf = 0) into Equation (1) leaves us with an equation involving k and the initial bow stretch xi. Solving for the spring constant k, we find

We are given the final speed of the arrow vf. To find the spring constant k, we need to also know the amount the bow stretches xi, but this information was not given. Relying on intuition and experience, we estimate xi ≈ 0.1 m. A value of x = 0.01 m = 1 cm seems much too small, whereas x = 1 m is too large. Our estimate is in the middle. Evaluating Equation (2) for k then gives

What did we learn?

Notice how we followed the general steps in our strategy for dealing with conservation of energy problems. We began with a picture and identified the initial and final states of the system (bow plus arrow). We then wrote the conservation of energy relation, Equation (1), and solved for the quantity of interest (k).

Figure 6.25B shows bar charts for the initial and final kinetic and potential energies. The initial potential energy is converted into the final potential energy, and the total energy KE + PE is conserved.