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Illustration 8.1: Force and Impulse
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So what do we mean by a force? Newton considered a net force as something that caused a time rate of change of momentum, Δp/Δt or dp/dt. However, C.D. Broad (Scientific Thought, 1923) wrote, "It seems clear to me that no one ever does mean or ever has meant by 'force', rate of change of momentum." So if Newton's statement seems odd it is because you are used to a special—and famous—case of Newton's general statement of the second law, that of Σ Fnet = ma. Restart.
Consider the force applied by the hand over a small Δt (this happens automatically at t = 1 s). Notice the change in momentum (position is given in meters and time is given in seconds). The arrow represents the change in momentum. Initially the mass of the cart is 1 kg. Change the mass to 2 kg. Does the change in momentum differ? No! But what does change is the final velocity; it is half of the velocity when the mass was 1 kg. The same force results in the same change in momentum in the same time interval.
Another way to represent this is in terms of the integral (the area) under a force vs. time graph. Check the box to see this graph. This area is called the impulse, which is a fancy name for Δp. What can you say about the impulse received by the cart, independent of its mass? Check the second box to find out. Again, it should be, and is, the same.
Consider the animation with the force applied by the hand over a large Δt (this happens automatically at t = 1 s). The difference between the animations is that in large Δt the force acts for a longer time and therefore the force causes a larger change in momentum. Again, the arrow represents the change in momentum, which is larger than the small Δt case.