0.11 Phy1080: motion -- variable velocity and acceleration  (Page 6/19)

Analysis of the output

You should be able to tell that your curve begins at a height of 6 feet and increases on a point-by-point basis out to about 3 seconds. The maximumvalue should occur around 3 seconds and the plot should go to zero between 6.25 seconds and 6.5 seconds.

Not a trip to outer space

As you are probably aware, shooting an arrow upward does not cause the arrow to go into outer space, unless the arrow is self-propelled and manages to reach avelocity commonly known as the "escape velocity."

Instead, for any practical value of initial velocity (neglecting air resistance), gravitationalattraction will eventually cause the arrow to slow down, reverse course, and fall back to earth with continually increasing velocity until it strikes theearth and stops. That is the message conveyed by the plot on your graph board.

The maximum height

At about 3 seconds and a height of about 161 feet, the kinetic energy provided by the initial velocity will have been dissipated by the gravitationalattraction of the earth. At that point, the arrow won't go any higher. Instead, it will start falling back toward the earth. Somewhere around 6.25 seconds, itwill strike the earth.

The shape of the curve

The shape of this curve is controlled by only two factors: the initial velocity of the arrow and the gravitational attraction of the earth. The archerhas some degree of control over the initial velocity, but has no control over the gravitational attraction of the earth.

In theory, in the absence of an atmosphere, if the arrow is shot straight up, the arrow should land in the same place from which it was shot. In practice in the real world, wind and other factors wouldprobably prevent that from happening.

Variable velocity

We learned in an earlier module that if velocity is uniform, the displacement should be the same during each successive equal interval oftime. However, we can see from Figure 4 that is not true in this case. For example, Figure 5 shows the displacement versus time for the first five time intervals and we can see that it is anything but uniform.

Successively decreasing or increasing displacement

Referring back to Figure 4 (along with Figure 5 ), we see that the displacement during the first 0.25 second time interval was 24 feet, but the displacementbetween 2.75 seconds and 3.0 seconds was only 1.9 feet, Furthermore, the displacement was -0.2 feet during the interval between 3 seconds and 3.25 second indicating that the arrow had begun falling back to the earth.

From that point until contact with the ground, the displacement increased during each 0.25 second interval.

In this case, the velocity clearly wasn't uniform. Instead it was variable.

Measuring variable velocity in the real world

Measuring variable velocity in the real world can be difficult. What you need to do is to measure the displacement of an object in as short a time interval aspossible and then divide the displacement by the value of the time interval. If you can do this at enough points in time, you can construct curves thatrepresent the variable velocity to which the object is being subjected.