I call the formula we develop here the freefall formula, but it really works for any smooth acceleration from an initial velocity of zero, so it really might better be called the "smooth acceleration from zero" formula, but that's too long.
We begin by considering an object dropped from some height, d (for distance). Its initial velocity is zero. We'll let downward motion define the positive direction, and the acceleration of gravity is the only acceleration we need.
We begin with the definition of velocity, and rearrange it to find distance:
Note that the velocity in that equation is always the average velocity, that's why it's written as v. Now the average velocity of any moving object (with constant acceleration) is just the final velocity plus the initial, divided by 2, the so-called arithmetic average.
Further, because our initial velocity is always zero for a dropped object (we assume we aren't throwing it down), we have:
That's the first part. Hold that thought for a while and we'll work from the other side, using the definition of acceleration:
Here we've substituted g (g = 9.8 m·s-2) for a because we're talking about freefall. Now recognizing once again that the initial velocity is zero, we have:
If we rearrange this equation, solving for vf, we get vf = gt. We can then substitute this back into our equation d = ½vf·t from above:
and that's the freefall formula: The distance fallen is equal to ½ the acceleration due to gravity multiplied by the square of the time of the fall,
The freefall equation can be used in any situation where the initial velocity is zero and the acceleration is constant.
Solution: To solve this problem we'll need to rearrange the freefall formula to find the time. Here's the process:
Now this problem is easy. We know the distance, 100 m, and the acceleration of gravity, g = 9.8 m·s-2 :
Strictly speaking, we should have a ± in front of that square root, but there is no such thing as negative time, so we don't bother with it.
Solution: "Near the surface of Earth" in this problem is just another way of saying "the acceleration due to gravity is g = 9.8 m/s2." It's less as we move away from the surface. This question is just a straightforward application of the freefall formula.
The first calculation is
The object (absent any air resistance) will fall 4.9 meters in the first second.
Here are the results of the identical calculation for 2, 3, and 4 seconds of freefall.
Notice that the results are quadratic. That is, they scale as the square of the time. The distance vs. time graph (i.e. distance on the y-axis, time on the x – it's always "y vs. x") would be half of a parabola.
Solution: The freefall formula works in any constant acceleration problem in which either the beginning or ending velocity is zero. That's just a consequence of how we derived it: Remember that we set the starting velocity to zero to begin with.
We should probably first convert those miles and hours to SI units first:
You could do those unit conversions in other ways; those are just the conversions I can remember. You'll develop your own set that will work fine. I like to sketch out a kind of time line diagram for problems like these so I can keep track of what I know and don't know. Here's one:
Now we want the acceleration, so we'll need a time. We can get that by knowing the distance and the average velocity:
Let's update our diagram:
Now calculate the time it took to brake over the 37.8 m distance:
Now we have all of the information we need to calculate the acceleration. We can either use the definition of acceleration or the freefall formula:
The definition of acceleration gives the same result:
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