# How to Calculate the Velocity of a Falling Object Using Calculus

Calculus > Velocity of a Falling Object

What’s the velocity of a falling ball? Image: By MichaelMaggs |Wikimedia Commons

Differential calculus can be used to calculate the velocity of a falling object, thanks in part to Galileo. Galileo discovered that the distance traveled by a falling object is proportional to the square of the time the object has been falling. Thanks to that discovery, plus a little differential calculus, the velocity of a falling object can be calculated in just few steps.

## Velocity of a Falling Object: Sample Problem

Sample problem: Frustrated with your calculus class, you throw your textbook out of your dorm window, which is 200 feet above your car in the parking lot. The height of the book, in feet over the car after t seconds is given by the function h(t) = 200 – 16t2. The book will dent your car if it’s going more than 100 feet per second. Will your car get dented?
Hint:The given equation is not for the velocity of a falling object. It is a position function.

Step 1: Differentiate the position function, h(t) = 200 – 16t2 to get the velocity function (you need to know the velocity to answer the question). 200 is a constant, so it disappears. 16t2 can be differentiated using the power rule. So the differentiated function is 2(16)t2-1 = -32t.

Step 2: Solve the position function for zero (in other words, when the height is zero) to find out when the book will hit the car. You know the velocity function from Step 1, so now you need to know what time the book will hit the car in order to use the function. Setting h(t) = 0 gives:
0 = 200 – 16t2
t2 = 200/16 = 12.5
t = 3.54

Step 3: Insert your answer from Step 2 into the velocity function from Step 1:
v(5) = -32(5)
v(5) = -113.28
The velocity is 113.28 feet per second when the book hits the car, which is more than 100 feet per second. Yes, there will be a dent!
That’s it!

Tip: A negative sign in a velocity equation indicates the height is decreasing.

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