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A hockey player received the puck and checked it. Then with a quick (approximately 25 centimeter) sweep of his stick, he slapped the puck across ice into the net. Our system, is the 0.2 kilogram puck received at low speed. Our event is its "slap" to goal which changes its speed to 30 meters per second.
Calculate the work of the "hockey-stick" that accomplished the shot.
♦ The puck moving horizontally on ice has constant potential energy. The energy of the "shot" becomes increased kinetic energy of the puck. The extrinsic energy equation has two terms, ΔKe and the work. We calculate the kinetic energy change. Ignoring losses to a "sloppy" slap shot, splintering of the stick... we obtain the LEAST work.
Calculate the average force applied to the puck by the stick.
♦ One purpose of this example is provoke an evaluation or approximation of the work integral. We already know the second kinetic energy, so we use that and write the energy equation as:
The integrand of the integral is the vector force, F, which probably varies in direction and magnitude during its displacement. The differential of the integral is a vector differential. To proceed we write the integrand and the differential in their magnitude-direction forms.
The direction of the force is written, eF and the direction of the differential displacement is written, es. With a clean motion of the stick, for the duration of the force, it is reasonable to expect eF = es so that eF·es = 1. The magnitude of the force is a function of the scalar distance along the path of applied force, s. The magnitude of the vector displacement differential becomes a measure of arc length.
Our resulting integral requires specification of how the force magnitude |F(s)|, varies in magnitude along the path of its application denoted by the length s (initially zero and finally 25 centimeters). This we do not know. So we apply the mean value theorem (MVT) to retrieve the average force.
It is often in analysis that you will encounter an integral with a "stubborn" integrand. Whenever this happens, we "give it a slap" by using the Mean Value Theorem of calculus.