Need help on this please

Question

A batter hits a fly ball. A scout in the stands makes the following observations.

 

Time(seconds).75, 1.5, 2, 2.75, 3.25, 4.75

 

Height(feet) 77, 133, 160, 187, 194, 169

 

circle the type of function that best models this data.

Linear; exponential; quadratic; other

 

use a graphing calculator to perform the regression for the best fit equation. Right the resulting equation below, rounding to the nearest hundredth.

 

Answer 1

Hello Hanna,

 

First, ANY ballistic object (thrown, hit, or otherwise launched objects under the influence of gravity are ballistic objects) describes a parabolic arc, which is given by quadratic equations (2nd-order polynomials).

 

Since this is a math class, you're probably not allowed to know physics. But it turns out the general equation for the height of a ballistic object is given by the general equation

 

h(t) = h₀ + v₀(t) - (1/2)gt²,

 

where h(t) is the height at time t, h₀ is the height at impact (approximate this as zero), v₀(t) is the initial upward velocity times duration of ball flight t (and would be the height of the ball in absence of gravity), and -(1/2)gt² is the downward acceleration of the ball under the gravitational constant, g, which is 32 ft/s² on Earth.

 

If you don't own a graphing calculator, get one by Texas Instruments or Hewlett-Packard. Something tells me you should've already done this, as the teacher is assigning problems that instruct you to use one.

 

In absence of such a calculator, enter the data on an Excel spreadsheet. (ANY personal computer running Windows will have three Microsoft Office applications on it: Word, Powerpoint, and Excel.) Before plotting the data, add the point 0 seconds, 0 feet to your data. Use time as your x-axis and height as your y-axis. The data will plot a downward-curving parabola, which confirms a 2nd-order polynomial equation: in other words a quadratic equation.

 

Excel's "add a trendline" drop-down menu pops up if you right-click on one of the data points. Choose "polynomial" as your trendline type. Select Order = 2. (The "order" of a polynomial is the exponent on the variable with the highest exponent in the equation.) Select "Set intercept = 0" (if you haven't already added (0, 0) to your data), "Display equation on chart," and, for giggles, "Display R² value on chart."

 

It will spit out your equation which paradoxically contains an acceleration term of -16.27 ft/s². This is a touch stronger a gravitational pull than we have here on Earth (we'd expect -16.08 ft/s²), but hey, you've got to work with the data you've been given.

 

Make sure to "write" the equation down. I don't know how you'd "right" it, unless you corrected the acceleration term to -16 ft/sec². But then your curve wouldn't fit your data!

 

Note that the equation Excel spits out for you, assuming you've used a 2nd-order polynomial fit, is in the form of the general equation given above. The R² = 0.9999 means that the data are pretty much ideal for that fit.

 

By the way, this ball doesn't even start coming down until at least 3.25 seconds have elapsed. Assuming it's got some distance to it as well, it's probably out of the park!

 

Get that calculator, Hanna! (They have them at Walmart cheap -- an HP 50g is right at $90, and the TI 83 Plus is $96, and these handheld calculators are so versatile that the user's manual for mine is over 850 pages long!)

 

 

Regards,

 

Eric Moline

McMinnville, OR

Answer 2

It is definitely not an exponential as the ball increases in height and then decreases in height. So it is a quadratic equation.

As far as performing the regression to find the best fit equation, my personal choice is Microsoft Excel. Plot it on a scatter plot then add a trendline (Quadratic) to get an equation.

Doing this, gives an equation of:

y= (-15.90)x² + (110.48)x +(3.02)

after rounding to two decimal places.

 

Hope this helps,

Chris