The goal of the experiment is to study the relationships between displacement, velocity and acceleration in a freely falling body, and to determine the acceleration of gravity.
EQUIPMENT photogate w/interface, picket fence, sponge rubber.
v(t) = vo - g t
y(t) = yo + vo t - g t2
The one-dimensional motion of a body under the influence of gravity alone is referred to as "free fall". This is an idealized situation where all other forces, such as air resistance, are neglected.
Figure 05-1: Plot of Vertical Position vs. Time for Freely Falling Body
When it undergoes free fall, a body experiences constant acceleration- the acceleration of gravity g.
From 1-D kinematics, we know that the instantaneous velocity is the slope of the position vs. time plot. From the position equation, we expect y(t) to increase as a quadratic function of the time. The plot of y(t) vs. t will be a parabolic curve, as illustrated in Figure 05-1. A slope is defined only for a straight line, so how is it possible to determine the velocity as the slope of y(t) vs. t?
The answer is to pick a point on this curve that will correspond to a specific time and y-value, and then to construct the tangent line to the curve at this point. The tangent line is the unique straight line that touches the curve a one and only one point. The slope of the tangent line at this particular point is therefore the instantaneous velocity at this point, which is negative in the case of free fall. It is clear from the figure that the velocity changes constantly, and that it is more negative (larger magnitude) at time t2 than it is at time t1.
Similarly, by looking at the velocity equation, we expect v(t) to decrease linearly with time. This is another way of saying that v(t) vs. t should be a straight line with a negative slope. The slope of the velocity plot is minus the acceleration of gravity, as illustrated in Figure 05-2.
Figure 05-2: Velocity vs. Time for a Freely Falling Body
It is possible to make measurements of the varying instantaneous velocity by constructing an object that has a number of relatively small, equally spaced divisions on it, and by measuring the time it takes for each division to pass by a fixed point.
This idea is implemented by using a photogate and special ruler called a "picket fence". The picket fence has a number of equally spaced alternating opaque and transparent bands that allow the photogate beam to be alternately blocked and passed and then detected. When connected to the computer, a series of time intervals Dt, can be recorded, one for each passage of the distance between bands Dx. The average velocity over each interval is obtained by taking the ratio:
In a similar way, the average acceleration over an interval can be determined by taking the difference between two successive velocity values Dv and dividing by the time interval Dt: