The force imparted on an object is equal to the change in momentum divided by the time interval over which the objects are in contact.

**Internal forces** are forces for which both Newton’s Third Law force pairs are contained within the system. For example, consider a two-car head-on collision. Define the *system* as just the two cars. In this case, internal forces include that of the fenders pushing on each other, the contact forces between the bolts, washers, and nuts in the engines, etc.

**External forces** are forces that act on the system from outside. In our previous example, external forces include the force of gravity acting on both cars (because the other part of the force pair, the pull of gravity the Earth experiences coming from the cars, is not included in the system) and the forces of friction between the tires and the road. If there are no external forces acting on a system of objects, the initial momentum of the system will be the same as the final momentum of the system. Otherwise, the final momentum will change by \begin{align*} \Delta \vec{p} = \vec{F}\Delta t\end{align*}. We call such a change in momentum \begin{align*} \Delta \vec{p} \end{align*} an **impulse**.

\begin{align*} \Delta \vec{p} = \vec{p_{f}} - \vec{p_{i}} \end{align*}

\begin{align*} \Delta \vec{p} = \vec{F_{net}} \Delta t \end{align*}

### Review

- You jump off of the top of your house and hope to land on a wooden deck below. Consider the following possible outcomes:
- You hit the deck, but it isn’t wood! A camouflaged trampoline slows you down over a time period of \begin{align*}0.2\end{align*} seconds and sends you flying back up into the air.
- You hit the deck with your knees locked in a straight-legged position. The collision time is \begin{align*}0.01\end{align*} seconds.
- You hit the deck and bend your legs, lengthening the collision time to \begin{align*}0.2\end{align*} seconds.
- You hit the deck, but it isn’t wood! It is simply a piece of paper painted to look like a deck. Below is an infinite void and you continue to fall, forever.
- Which method will involve the greatest force acting on you?
- Which method will involve the least force acting on you?
- Which method will land you on the deck in the least pain?
- Which method involves the least impulse delivered to you?
- Which method involves the greatest impulse delivered to you?

- You punch the wall with your fist. Clearly your fist has momentum before it hits the wall. It is equally clear that after hitting the wall, your fist has no momentum. But momentum is always conserved! Explain.
- You look up one morning and see that a \begin{align*}30 \;\mathrm{kg}\end{align*} chunk of asbestos from your ceiling is falling on you! Would you be better off if the chunk hit you and stuck to your forehead, or if it hit you and bounced upward? Explain your answer.
- A baseball player faces a \begin{align*}80.0 \;\mathrm{m/s}\end{align*} pitch. In a matter of \begin{align*}.020\end{align*} seconds he swings the bat, hitting a \begin{align*}50.0\;\mathrm{m/s}\end{align*} line drive back at the pitcher. Calculate the force on the bat while in contact with the ball.
- A place kicker applies an average force of \begin{align*} 2400 \;\mathrm{N}\end{align*} to a football of \begin{align*}0.40 \;\mathrm{kg}\end{align*}. The force is applied at an angle of \begin{align*}20.0\end{align*} degrees from the horizontal. Contact time is .\begin{align*}010\end{align*}sec.
- Find the velocity of the ball upon leaving the foot.
- Assuming no air resistance find the time to reach the goal posts \begin{align*}40.0 \;\mathrm{m}\end{align*} away.
- The posts are \begin{align*}4.00 \;\mathrm{m}\end{align*} high. Is the kick good? By how much?

- Your author’s Italian cousin crashed into a tree. He was originally going \begin{align*}36 \;\mathrm{km/hr}\end{align*}. Assume it took \begin{align*}0.40\end{align*} seconds for the tree to bring him to a stop. The mass of the cousin and the car is \begin{align*}450\;\mathrm{kg}\end{align*}.
- What average force did he experience? Include a direction in your answer.
- What average force did the tree experience? Include a direction in your answer.
- Express this force in pounds.
- How many g’s of acceleration did he experience?

- Serena Williams volleys a tennis ball hit to her at \begin{align*}30\;\mathrm{m/s}\end{align*}. She blasts it back to the other court at \begin{align*}50 \;\mathrm{m/s}\end{align*}. A standard tennis ball has mass of \begin{align*}0.057 \;\mathrm{kg}\end{align*}. If Serena applied an average force of \begin{align*}500\;\mathrm{N}\end{align*} to the ball while it was in contact with the racket, how long did the contact last?

### Review (Answers)

- .
- .
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- a. \begin{align*}60 \;\mathrm{m/s}\end{align*} b. \begin{align*}.700 \;\mathrm{sec}\end{align*} c. yes, \begin{align*}8.16 \;\mathrm{m}\end{align*}
- .
- a. \begin{align*}11000 \;\mathrm{N}\end{align*} to the left b. tree experienced same average force of \begin{align*}11000 \;\mathrm{N}\end{align*} but to the right c. \begin{align*}2500 \;\mathrm{lb}\end{align*}. d. about \begin{align*}2.5\end{align*} “g”s of acceleration
- a. \begin{align*}0.00912 \;\mathrm{s}\end{align*}