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PURPOSE

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 due to gravity.

EQUIPMENT  photogate w/ DataLogger interface, picket fence.

RELEVANT EQUATIONS

DISCUSSION

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. When it undergoes free fall, a body experiences constant acceleration: the acceleration due to 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 decrease as a quadratic function of the time. The plot of y(t) vs. t will be a parabolic curve, as illustrated in Figure 8-1. 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 at 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 shown. It is clear from the figure that the velocity changes continuously, 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 the v(t) vs. t should be a straight line with a negative slope. The slope of the velocity plot is the acceleration due to gravity, as illustrated in Figure 8-2.

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 alternately be blocked and transmitted.

When connected to the computer, a series of time intervals, Δt, can be recorded, one for each passage of the distance between bands Δx. The average velocity over each interval is obtained by taking the ratio:

(1)

for each interval. An approximate plot of the instantaneous velocity can be made by assuming that the distance interval is relatively small, and that the time instant when the v(t) value is plotted is midway between the ends of the Δt interval. In this approximation, we take the average velocity over the small interval to be a good estimate of the instantaneous velocity in the middle of the interval.

In a similar way, the average acceleration over an interval can be determined by taking the difference between two successive velocity values Δv and dividing by the time interval Δt:

(2)

Again, the approximation is that if the interval is sufficiently small, the average acceleration becomes a good estimate of the instantaneous acceleration at the midpoint of the interval.