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Question

Dr. Peabody shot a rocket off the roof of a 42-foot tall building. After three seconds, the rocket reached its highest point of 96 feet.

Model this situation with a quadratic function, and use the graph to determine how many seconds it will take the rocket to hit the ground?

Answer 1

Alexis, once the rocket has reached the top of its flight, it is simply a falling body. We can use the falling body equations to learn the relationship between time and height.

let h(t) be the height of the rocket t seconds after it reached the top. So h(0) = 96.

The formula is h(t) = 96 -1/2 at², where a = 32 ft/Sec/Sec

We want to find the t that makes h = 0

h=96-16t²

t² = -(h-96)/16 where h = 0

t = sqrt( 96/16 )

Finish the calculation, and you’ll have the number of seconds after the rocket reached its peak.

NOTE: the other answers to this question all come up with 7 seconds from launch or 4 seconds from the apex. Here’s why I disagree:

This is partly a physics question and partly a math question. A rocket has fuel which it burns up as it goes up, so it has a varying acceleration and, since it consumes the fuel, it has a varying mass. As a result, its flight upward is not a parabola. We don’t know its complicated formula. But... we don’t need to!

Once it has reached its apex and run out of fuel, it is simply a falling object, and its behavior follows the quadratic function given above, and the answer is

sqrt(96/16) = sqrt(6) after reaching the apex, or

3+ sqrt(6) after launch.