Calculus > Velocity of a Falling Object
How you solve for the velocity of a falling object depends upon what information you’ve been given. For example, if you’ve been given a time (usually in seconds), then the velocity of any falling object can be found with the equation v = g*t, where g is acceleration due to gravity. However, if you’ve been given a position function (usually for the height), then you need a little calculus to derive the answer.
- Velocity of a Falling Object: v = g*t
- Velocity of a Falling Object With Calculus: The Position Function
- Other Useful Equations
A falling object is acted on by the force of gravity: -9.81 m/s2 (32 ft/s). Gravity will accelerate a falling object, increasing its velocity by 9.81 m/s (or or 32 ft/s) for every second it experiences free fall.
In order to find the velocity of a particular falling object, just multiply time (t) by gravity (t). The formula is:
v = g*t
v = -9.81 m/s2*t
Example #1: An object falls for 1.2 seconds. What is it’s velocity?
- v = -9.81 m/s2*t
- v = -9.81 m/s2*1.2s
- v = 11.77200 m / s2
Calculus is very useful for finding the velocity of a falling object if all you have is a position function, like the height of an object. First, differentiate the position function to get the velocity function. Then use that function to find the answer.
Example #2: Position Function
The velocity equation v(t) is the derivative of the position equation. If you’re given a position equation like h(t) or s(t), you’ll need to differentiate that function in order to find the velocity of the falling object.
Example 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.
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.
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!
Tip: A negative sign in a velocity equation indicates the height is decreasing.
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