\(\def \u#1{\,\mathrm{#1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \tau{\uptau}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\) \(\def \ccw{\circlearrowleft}\) \(\def \cw{\circlearrowright}\)
Chapter 5: Impulse and Momentum
1.

Impulse

Impulse
\(\vec J\)
Ns

Forces are usually applied over a period of time. I push a box for 10 seconds, or a baseball bat makes contact with a ball for 5 microseconds, and so forth. The longer a force is applied to an object, the bigger an effect that force will have. This effect is quantified by the force's impulse on the object, which is given by the formula

$$\vec J=\vec F_{avg}\Delta t$$

where $\Delta t$ is the time over which the force is applied, and $\vec F_{avg}$ is the average force during that time. Impulse is a vector and use units of Newton-seconds (Ns). A large impulse may be created by a small force over a long period of time, or by a large force over a short period of time.

Example

(no alternate text)

For example, a falling ball of mass 2kg feels a force $mg$ of about 20N downward. If the ball falls for 8 seconds, then gravity exerts an impulse of $\vec J=160\u{Ns}$ downward on the ball.

The collision here takes less than a millisecond, but the forces are so large (kilonewtons) that the overall impulse (the area under the curve) is about 10 Ns.

In the case of a collision, like a baseball bat hitting a ball, the force is usually not constant: it starts off small, increases to a maximum as the ball comes to a stop, and then decreases after the ball reverses direction. This is why the impulse is defined in terms of the average force. If one can create a graph of the force as a function of time, then the impulse of the force is equal to the area between the curve and the x-axis.