The Hit
Graphing Calculator Activity

Bat Type Activity
Bat Length Activity
Coefficient of Restitution (COR) Exploration
Graphing Calculator Activity

Advanced Activity

Calculator skills finding the maximum of a function and the area under a curve

Directions are based on the use of a TI83 graphing calculator.

Momentum is a physicist's best measure of motion it's the essence of motion because it tells how much is moving, how fast it's going and where it's going, all in just two little symbols, m and v. The product of the mass, m, and the velocity, v, is defined as momentum. Isaac Newton (hero of fans of both physics and baseball) explained that force is the rate of change of momentum, or the change in momentum over time.

F = change in (mv) / T

If we're talking about baseball, the time, T, is the contact time between the bat and the ball, which is typically about 0.7 millisecond (ms), or 0.0007 second. The mass of a baseball is 145 grams, or 0.145 kg. Using the speeds (above) of a typical pitch and a hard liner, the change in velocity is 40.2 m/s - (-49.1 m/s) or 89.3 m/s. (Do you see why the change in velocity is not just 6.9 m/s?)

The change in momentum, then, is (0.145 kg)(89.3 m/s) = 12.95 kg-m/s. That doesn't seem like so much, but since the contact time is so small, the force is

F = 12.95 kg-m/s / 0.0007 s
= 18,500 N

Wow, more than 18,000 N! Since one pound is the same as 4.45 N, that's a force of about two tons! No wonder you hear such a loud CRACK! from the impact.

But wait, there's more! That number is the average force during the contact time. The force changes during the contact in a sort of bell-shaped curve given by this equation:

where F_(t) means the force as it changes over time, T is the contact time between the bat and ball, t is the elapsed time, and ()v) means the change in velocity.

To graph that equation, first be sure your calculator is in RADIAN mode. Don't know how to check that? Go to the mode button next to the 2nd key.

Enter this equation in your function editor to graph the example we've considered so far, where the ball of mass 0.145 kg changes its velocity by 89.3 m/s in just 0.0007 second:

y1 = 2*.145*89.3/.0007*(sin(Bx/.0007))²

Press [WINDOW] to set the viewing rectangle. Since x is time, set x_min = 0 and x_max = .007. The variable y represents force, which will range from zero to about twice the average force, so let y_min = 0 and y_max = 40000. Press [GRAPH] and have a look. Pretend you are watching the force between the bat and the ball in super slow motion. The actual impact moves along about ten thousand times faster than we see it on the screen!

Going Further

1. Just how big is that maximum force? Use the CALC function that's [2^nd][TRACE] and choose item 4 from the menu. Use the left arrow key to move the cursor anywhere to the left of the peak and press [ENTER]. Then move the cursor anywhere to the right of the peak and press [ENTER] again. Press [ENTER] once more, and the screen will show the coordinates of the maximum force. Remember that, in our example, x is time and y is force, so we see that the maximum force of about 37,000 N occurs about halfway through the swing. Remember, too, to round severely the numbers that you see, because we know that the contact time is only accurate to one significant figure, but the calculator doesn't know about that limitation in accuracy.
   
   Note that the maximum force is just twice the average force we calculated earlier. That's true because the force varies with time through a sine-squared function. If it had been a quadratic equation, for example, then the peak would not have been twice as high as the average. It seems that nature happens to use the sine-squared function for baseball! (See "Physics and Acoustics of Baseball and Softball Bats" at www.gmi.edu/~drussell/bats-new/impulse.htm.)

2. Want to see about the coolest thing your calculator can do? The area under the force vs. time curve is the product of the force multiplied by the time, which is called the impulse. It's numerically equal to the change in momentum. It's easy to calculate the area under a rectangle or triangle, but how can you find the area under a funky curve like this one?
   
   Your calculator can do it easily. Again, use the CALC function, but this time choose item 7 from the menu. The symbols you see there stand for the calculus function of integration, which is the way to find the area under any curve. When the calculator asks for the lower limit -- that means the starting time -- just press zero. For the upper limit (that is, the ending time), enter .0007. Then watch what happens when you press [ENTER]! After an amazing display, the bottom of the screen shows the total area under the curve, which for us is the impulse between the bat and the ball, which is also the change in momentum of the ball, about 12.95 kg-m/s.

3. Try some other combinations of pitches and swings. Estimate (that means, make up some reasonable numbers for) the speed of the pitch and the speed of the ball after the hit. Surprisingly, the contact time depends more on the elasticity of the ball than on how you swing, so don't change the contact time by more than a few ten-thousands. What do you find for the maximum force in each case?

4. Calculate the acceleration of the ball. We used Newton's second law earlier in the form where force equals the rate of change of momentum. More commonly, the second law is written as force equals mass times acceleration, or
   
   F = ma
   
   Divide the average force we found above (about 18,500 N) by the mass of the ball (0.145 kg) to find the acceleration. It's an amazing value of nearly 128,000 m/s², or about 12,800 times the acceleration due to gravity, or 12,800 g's! That much acceleration applied to your body would squish you like a bug on a windshield.