Timed Free Fall

One such experiment is shown here. It involves timing the free fall of a small steel ball. Because of the large density of steel and the small drag force on a sphere we can neglect the effects of air drag in this fall. We have a timing mechanism set up so that we can determine the time it takes to drop from rest a certain known distance. We have two objectives with this experiment

1. Establish that free fall is motion with constant acceleration
2. determine the acceleration of gravity common for all items falling close to earth in vacuum or when air-drag can be neglected.

We assume that this is fall with constant acceleration and derive a prediction for how the fall time should depend on the distance of the fall. For this we go straight to our master equation

Here you notice that I have replaced x by y as is customary when dealing with motion in the vertical direction. Here I want to make an extremely important point which pertain to problem solving. I asked you to establish symbolic variables for all physical quantities before you start setting up the physics equations. The other thing you need to make clear when dealing with position velocity and acceleration is the direction in which your x or y variable increases ie. the positive direction for your number line. A vast amount of errors in physics problems come about because the student did not make clear to himself the positive direction of the x-axis or y-axis.

In this problem I will choose up as the positive direction. I will denote the positive magnitude of the presumed acceleration of gravity by g. We expect this acceleration to be oriented downward in other words a is negative because it is opposed to the direction of increasing y.

Inserting in my master equation I have

We choose the origin of our y-axis to be at the end of the fall hence

where we have introduced h as the height of the fall. Furthermore we shall release the ball from rest and therefore

Inserting these results simplifies our equation to the point that we can easily solve for t:

If we choose to evaluate this equation at the time where y=0 we obtain
 
We will now run the experiment at two values of h to determine whether the free fall is consistent with this formula and therefore likely to indeed be motion with constant acceleration. We are set up for h=4 m and h=1 m. Since the heights differ by a factor 4 and h enters as a square root we should expect the times to differ by a factor 2. (Do the experiment)

We obtain the following times

The results are consistent with the prediction that free fall without air drag is motion with constant acceleration.

Now for our second point we determine the acceleration by solving Eq. 24 for g to get

We are pleased to see that the dimension of this expression is correctly . As good experimentalists we average the results of our two measurements:

This is very close to the accepted value of the acceleration of gravity which is 9.81 m/s.

We will not be able to solve more problems on this at lecture but I would like to emphasize the importance of you becoming very handy with the equations for motion with constant acceleration. Ask your TA at section to go through a bunch of these problems and make sure you fully understand the problem due next week about the astronaut.