Question

Neil Armstrong throws a ball down into a crater on the moon. The height s (in feet) of the ball from the bottom of the crater after t seconds is given in the following table:

(a) Find the average speed from t = 1 to t = 4 seconds.

(b) Find the average speed from t = 1 to t = 3 seconds.

(c) Find the average speed from t = 1 to t = 2 seconds.

(d) Using a graphing utility, find the quadratic function of best fit.

(e) Using the function found in part (d), determine the instantaneous speed at t = 1 second.

Short answer

Part (a) $-23\frac{1}{3}\mathrm{ft}/\sec$

Part (b) $-21\mathrm{ft}/\sec$

Part (c) $-18\mathrm{ft}/\sec$

Part (d) localid="1652236634358" $-2.63t^2-10.269t-999.993$

Part (e)localid="1652236624023" $-15.531\mathrm{ft}/\sec$

Step-by-step solution

Given information

The height s (in feet) of the ball from the bottom of the crater after t seconds is given in the following table.

Part (a) Step 1. Calculate average speed from t = 1 to t = 4 seconds.  

Use the average value formula to find the average speed between t=1 and t=4.

$\begin{array}{rcl}\mathrm{Average}\mathrm{speed}& =& \frac{\mathrm{s}\left(4\right)-\mathrm{s}\left(1\right)}{4-1}\\ & =& \frac{917-987}{3}\\ & =& -\frac{70}{3}\\ & =& -23\frac{1}{3}\mathrm{ft}/\sec \end{array}$

Part (b) Step 1. Calculate the average speed from t = 1 to t = 3 seconds. 

Use the average value formula to find the average speed between t=1 and t=3.

$\begin{array}{rcl}\mathrm{Average}\mathrm{speed}& =& \frac{\mathrm{s}\left(3\right)-\mathrm{s}\left(1\right)}{3-1}\\ & =& \frac{945-987}{2}\\ & =& -\frac{42}{2}\\ & =& -21\mathrm{ft}/\sec \end{array}$

Part (c) Step 1. Calculate the average speed from t = 1 to t = 2 seconds.  

Use the average value formula to find the average speed between t=1 and t=2.

$\begin{array}{rcl}\mathrm{Average}\mathrm{speed}& =& \frac{\mathrm{s}\left(2\right)-\mathrm{s}\left(1\right)}{2-1}\\ & =& \frac{969-987}{1}\\ & =& -\frac{18}{1}\\ & =& -18\mathrm{ft}/\sec \end{array}$

Part (d) Step 1. Find the best fit model.

By using a graphing utility,

The best fit model for the given table is,

$-2.63t^2-10.269t+999.993$

Part (e) Step 1. Calculate the instantaneous speed at t = 1 second. 

Calculate the derivative of the position function and substitute 1 for t.

$$\begin{gathered} s\left(t\right)=-2.63t^2-10.269t-999.993 \\ v\left(t\right)=\frac{ds}{dt}=-5.26t-10.269 \\ v\left(1\right)=-5.26-10.269 \\ v\left(1\right)=-15.529\mathrm{ft}/\sec \end{gathered}$$