Derivation of the Freefall Equation
We begin by considering an object dropped from a height. Its initial velocity is zero. We'll let downward motion define the positive direction. We begin with the distance formula, and note that the velocity in that equation is the average velocity. Realize that the average velocity of a falling object (with constant acceleration) is just the final velocity plus the initial, divided by 2:
Now use the definition of acceleration (here we'll use the acceleration of gravity, g = 9.8 ms-2) and the special fact in this problem that the initial velocity was zero, to get:
Finally, we put these two results together, substituting gt for vf:
... to get the freefall equation:
This animation is a simulation of 3 seconds of freefall, in which a falling object travels about 44 meters. The horizontal lines are placed at 0.1 second intervals. The vertical distance between each line and the top line (the beginning of the freefall) scales as the square of the elapsed time.
The acceleration of gravity is the same for any object at the surface of Earth (see gravity).
Nothing special about down
Note that there is nothing special about constant acceleration downward. We could easily use the freefall equation for constant acceleration in the horizontal direction. For example, to answer the question, how far does a cyclist travel if she accelerates from rest at a rate of 2 m·s-2 for 1 minute? ... we could just use the freefall equation, substituting 2 m·s-2 for g=9.8 m·s-2.
Be careful ...
Remember that we developed this formula starting with an object at rest (initial velocity = 0). It won't work for an object already in motion. For example, the freefall equation can't be used to directly answer a question like, "How far does an object already falling at 44 m/s travel in the next 2 seconds?"
Rearrangement of the freefall equation
We can also use the freefall equation to find time as a function of distance fallen. The steps for deriving the time equation from the distance equation are shown here.
Practice
- How far does an object fall after free-falling from rest for 8 seconds? How about 10 seconds?
- One World Trade Center in New York City is 541 m tall. How long would it take an object to fall from the top of the building if air resistance is negligible?
- A baseball is thrown straight upward with an initial velocity of 30 m/s. How far high will the ball go before it begins its downward trip? How long will the upward and downward trips take?
- A car, initially traveling at a constant velocity, accelerates at a rate of 1.0 m·s-2 for a period of 12 s. If the car traveled 190 m during this period, what was the velocity of the car when it began to accelerate?
- While traveling out in the country at 50 mi./h, your car's engine and brakes stop working any you coast to a stop in 25 seconds. Calculate your average acceleration during the time after the motor shuts off.
- A ball rolls down an inclined plane with constant acceleration of 2.5 m·s-2.(a) How fast is the ball rolling after 3s? (b) How far has the ball rolled in this time? (c) How far will the ball have traveled when its velocity reaches 15 m/s?
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