This is Zhang Juanjuan, immediately after applying an impulse to an arrow. Impulse is something which gets less airtime than work in freshman physics, but it’s nonetheless very important. It’s sort of momentum’s version of work.
You’ll remember from Monday that there are two main things to keep in mind about work.
1. Work is defined as force through a distance
2. Work equivalent to the change in energy
You can think of impulse the same way.
1. Impulse is defined as a force thorough a time
2. Impulse is equivalent to the change in momentum
Crucially, impulse is a vector and work is not. Force is a vector because it has a magnitude and it’s being applied in a particular direction. But work isn’t a vector because it doesn’t have a direction – it’s just an amount of energy. Mathematically speaking, the dot product of two vectors is not itself a vector, it’s just a number.
Impulse however is a force vector multiplied by elapsed time, which is just a number. A vector times a number is another vector. So when Zhang Juanjuan fires that arrow for the gold medal, she has done work (the force of the string through its distance of travel) and applied an impulse (the force of the string during the time it took to travel that distance).
Now let me get some mathematical justification out of the way. The force might not be constant, so the total impulse is the sum of each bit of force times each bit of time during which it acted. That means by definition impulse is
But via the definition of force as the change in momentum with respect to time, we have
Still with me? If so, you might be curious as to why anyone would bother with this concept. The main reason is that it allows you to not worry about the details of the force as long as you can say something about the change in velocity. For an easy example, a baseball batter hitting a ball has applied a tremendous force over a very small time and distance, so the details of the work are difficult to calculate. But the initial and final velocities of the ball are easy to see, and so you can work backwards to find the impulse and from there find something about the forces involved. You’ll also see an extension of this concept called specific impulse used in rocketry to quantify how much momentum change you can get out of a given quantity of fuel.
Just as work presages a rewriting of Newton’s force mechanics into an energy-based form, impulse can be thought of as a first step in writing Newton’s laws in their Hamiltonian form. The Hamiltonian itself is the total energy expressed in terms of a generalized version of momentum.