Work and the work-energy theorem
Work measures force delivered through distance, and all of it shows up as a change in speed.
The work formula
Work is force magnitude times displacement times the cosine of the angle between them. Only the force component along the motion does work: pushing perpendicular contributes zero. The angle matters silently in many problems, since a rope at an angle or a force that opposes the motion changes the sign or shrinks the contribution.
W = Fd\cos\thetaThe work-energy theorem
The net work done on an object equals its change in kinetic energy. Positive net work speeds it up, negative net work slows it, and zero net work leaves the speed unchanged even if individual forces did work. Kinetic energy is (1/2)mv^2, so the theorem ties force and distance directly to speed without ever mentioning time.
W_{\text{net}} = \Delta KE = \tfrac{1}{2}mv^2 - \tfrac{1}{2}mv_0^2Worked example
A 0.145 kg baseball arrives at 40 m/s and is stopped in the catcher's mitt. Its kinetic energy drops from (1/2)(0.145)(1600) = 116 J to zero, so the mitt does -116 J of work on the ball. If the ball squashes into the mitt over 0.10 m, the average stopping force is 116/0.10 = 1160 N backward, a force worth remembering next time someone offers to catch.
W = 0 - \tfrac{1}{2}(0.145)(40)^2 = -116 \text{ J}Flashcard
Work is force times displacement times the angle factor
Check your understanding
A rope pulls a sled 5.0 m across ice with a 10 N force at 60 degrees above the horizontal. How much work does the rope do?