Modern physics equipment includes a distance probe as well
as a force probe. The distance probe is an ultrasound radar. It
sends 40 KHz sonic waves at a set number of times (pips) per second,
and analyzes the reflections off objects. The distance probe can
be thought of as an electronic bat. Like any other radar, the
distance probe has certain limitations. It will not detect small
angular objects such as rolling dice, nor pointed objects such
as cones and knife edges. On the other hand, the distance probe
can differentiate between distances as small as a newspaper thickness.
The software that operates the distance probe calculates distance,
instantaneous velocity and instantaneous acceleration.
A basketball is dropped to the floor. The distance probe records the delay time between pulse emission and return, and simultaneously calculates the distance to the ball, its instantaneous velocity, and its acceleration.
The distance-time graph consists of many parabolas. These parabolas are simply the bounces of the ball. The peaks of the parabolas decay because the ball is loosing some of its kinetic energy to heat on every impact with the floor. This graph is flipped (up side down) because the probe is measuring the distance from the ball as positive.
Note: the initial height of the ball above the floor .was ... 0.76 meters.
Click the link under the graph to replace it with the velocity-time graph.
The main feature of the velocity-time graph is the set of parallel
lines across the entire graph. The positively sloped lines have
a slope of 9.8 m/s which
is the acceleration due to gravity. The acceleration is positive
because the distance probe is configured to record downwards as
positive. The time intervals for which the graph has a positive
slope are the bounces of the ball. When the positive sloped line
intersects the x-axis the ball is at the highest point and the
instantaneous velocity is zero.
The other set of lines in the graph are sharply sloped downwards
and are not parallel to each other. These lines are the time intervals
for which the ball is in contact with the floor. The velocity
changes rapidly from downwards to upwards during the impact on
the floor, so the acceleration (the slope) is large. The time
for which the ball is on the floor is independent of the violence
of the impact because the ball performs half a period of simple
harmonic motion!
The coefficient of restitution is the ratio of the velocities before and after the impact on the floor. A graph was drawn to establish a relationship between the coefficient of restitution and the violence of the impact. The coefficient of restitution is constant. Its value is 0.80 ± 0.05.
The new window shows the acceleration-time graph. Since f = ma , a force-time graph is obtained by multiplying the acceleration values by the mass of the ball which is 305 grams.
Click the link under the graph to replace it with the force-time graph.
The force-time graph consists of many peaks which are the impacts on the floor. The horizontal lines between the peaks are simply the free motion of the ball in the air. They are at zero force because no force is applied to the floor. As shown in the acceleration and force graphs, the duration of the floor impact is nearly constant. This is indicated by the constant 'width' of the lines. It can be shown that the average force on the floor after the ball is released is just the weight of the ball [see the Appendix]. T
he average impact is obtained by dividing the integral across the graph by the time - which is indeed equal to the weight of the ball!
The work here introduces acceleration-time graphs modified to become force-time graphs. The modification is done by copying the data points from the Mac Motion software to the Graphic Analysis software, manipulating the values of the points and returning them to Mac Motion. The manipulation can involve multiplying or adding a constant etc.
When the objects are large enough and the impact is slow enough,
using this method to obtain force-time graphs is more efficient
and accurate than using a force probe as in Chapter 1. In contrast
to the force probe, the interaction of the distance probe with
the apparatus is minimal. The distance probe can handle any acceleration
and does not have a force limit like the force probe.