| 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.
| Starting points: |
- Speed of a typical pitch = 90 mph
- One meter per second equals 2.24 mi/hr,
so
- 90 mph (1 m/s / 2.24 mph) = 40.2
m/s
- Speed of a hard line drive = 110 mph
- 110 mph (1 m/s / 2.24 mph) = 49.1
m/s
- Since the line drive is moving in the opposite
direction of the pitch, we'll call its velocity
negative, or -49.1 m/s
|
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 xmin = 0 and xmax
= .007. The variable y represents force, which will
range from zero to about twice the average force,
so let ymin = 0 and ymax = 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
-
Just how big is that maximum force? Use the
CALC function that's [2nd][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.)
-
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.
-
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?
-
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.
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