Velocity of a falling object

Example   Ever since you started your calculus class you've suffered from blinding headaches. Nothing helps. Acupuncture, drugs, counseling, you've tried them all, but the headaches get worse and worse. The pain is unbearable, and you decide to end it all. Your drive to the middle of the Golden Gate Bridge and climb over the safety rail, 400 feet above the water. With that pain, life is not worth living, so you fling your calculus text (which you carry everywhere) over the edge, and jump out after it. Your height in feet over the water after t seconds is given by the function h(t) = 400 - 16t².

a. How long till you hit the water?

Solution to a): We don't need to do any fancy differentiation for this part. We just need to solve for when the height is zero. Setting h(t) = 0 gives:

$$\begin{eqnarray*}0 = 400 -16t^2 \\ t^2 = 400/16 = 25 \\ t = 5 \end{eqnarray*}$$

So you hit the water after 5 seconds.

Right after you let go of the textbook your headache disappears. You realize that it is that hated text that has been the cause of all your pain. Suddenly life is a realm of wonder, calling for your presence.

b. You had lots of diving lessons when you were a kid. If you are traveling with a velocity of less than 200 feet per second, you can survive the plunge. Will you survive?

Solution to b): We need to calculate your velocity  when you hit the water. Differentiating h(t) to get the velocity gives

v(t) = h'(t) = -32t.

So the velocity when you hit the water, when t=5, is given by

v(5) = -160.

You hit the water at a speed of 160 feet per second, so you'll make it. The minus sign in the velocity indicates that the height is decreasing. Now, if you can just swim 2 miles in freezing water against a fierce current, you may still be able to make your 3 PM calculus class.