Mathematics for the liberal arts (2013)

Part I. MATHEMATICS IN HISTORY

Chapter 4. CALCULUS

4.2 Average and Instantaneous Velocity

A good way to begin learning calculus is to think about falling objects. Consider the simple question: A pencil falls from a four foot high counter; how fast does it hit the floor?

To answer this question, we might create an experiment where we carefully time a pencil as it falls. If we did that, we would find that it takes the pencil almost exactly one-half second to fall four feet. Knowing that rate is distance divided by time, we might then conclude that the speed of the pencil is

In practice, we try to be careful to measure distance by subtracting earlier positions from later positions. Since the height of the pencil at time zero was 4 feet and the height of the pencil after one-half second was 0 feet, we actually get the speed of the pencil to be

We use the word velocity to describe speeds when direction is important. In this case, the velocity is negative to indicate that the pencil is falling (i.e., moving in a downward direction).

A little more thought will convince us that this is not the right answer. Dividing total distance by total time gives us the average rate, the average velocity of the pencil. The pencil is barely moving when it first starts falling and then goes faster and faster under the influence of gravity. The average velocity (being somewhere in the middle) will be faster than the very slow speed at the beginning of the pencil’s descent and too slow to be the speed at which the pencil hits the floor.

Perhaps, if we had a quick eye, we could see where the pencil was after 1/4 seconds. If we had a very quick eye, we would know that after 1/4 seconds the pencil was still about three feet above the floor. Since it falls the last 3 feet in only 1/4 seconds, we could redo our calculation to get

This is better, but it is still too slow for the same reason that our first estimate was too slow (because we want the velocity at the bottom, not somewhere in the middle of the descent).

What if we could continue making estimates over shorter and shorter time intervals? Over a very short interval, there isn’t much time for the pencil to accelerate, so the average velocity and the velocity we want should be almost exactly the same.

It would take a fast eye to see where the pencil is 1/16 or 1/32 of a second before the pencil hits the floor, but a video camera takes pictures at a frame rate of one frame every 1/29.97 ≈ 0.03337 seconds. If we dropped our pencil on camera, we could flip through frames and discover these data points:

Graphically, the data look like Figure 4.1.

Figure 4.1 A graph of pencil heights over time.

In 1638 Galileo asserted that the distance fallen is proportional to the square of time. Thus, the distance an object falls has an equation of the form d = kt², where k is a constant. Because our data points show how high the pencil is (starting from a 4 foot counter), adapting Galileo’s idea, there should be a formula for our data that looks like y = 4 – kt². Indeed, you can check by hand that these points are very closely predicted by the formula

This is the formula for the curve shown connecting the dots on the previous plot.

Once we know this formula, we can be as precise as we like about estimating the speed the pencil hits the floor. For example, the average speed during the last 1/16 seconds would be

We could continue to use shorter and shorter intervals, but that would mean repeating essentially the same computation over and over. We can save ourselves a lot of tedium by introducing a small bit of algebra. Let h be the length of the time interval we want to use (it could be the last 1/4 seconds, the last 1/16 seconds, or even smaller). Then the average speed of the pencil from time 0.5 – h to time 0.5 seconds is

It is now quick work to calculate as many estimates of the pencil’s collision speed as we like.

As h gets smaller and smaller, it is apparent that –16 + 16h gets closer and closer to –16. This gives us the true answer to our question. Over any time interval, the average velocity of the pencil will be some number larger (i.e., less negative) than –16 ft/s. But as the time intervals become shorter and shorter, the average velocities get closer to –16 ft/s, the speed of the pencil the instant it hits the floor.

There was nothing particular about the time t = 0.5 except that it happened to be the time when the pencil hit the floor. We could just as easily have estimated the speed of the pencil at any other instant during the descent. For example, let’s check the speed of the pencil the moment it starts to fall, at time t = 0, by finding the average speed over the time interval [0, h].

In this case, as h gets smaller and smaller (closer and closer to zero) the average speed goes to 0. This agrees with our intuition. Just for an instant when the pencil starts falling, it isn’t moving at all. Its instantaneous velocity at time 0 is 0 ft/s.

EXERCISES

4.1 Find the instantaneous velocity of a pencil dropped from a height of 4 ft when t = 0.25 s, i.e., the moment its height is 3 ft.

4.2 An object dropped from a height of 16 ft takes approximately one second to strike the ground, and it has a height function of y = 16 – 16t².

a) Determine the velocity of the object as it hits the ground.

b) Determine the velocity of the object at time t = 0.5 seconds.

c) Determine the velocity of the object at time t = 0.25 seconds.

d) Determine the velocity of the object at time t = 0 seconds.

4.3 A more precise height function for a pencil dropped 4 feet is y = 4 – 16.1t². It only takes about 0.49844 seconds (not a full half second) for the pencil to reach the floor.

a) Find the velocity at which the pencil strikes the floor using this height function and impact time.

b) Find the velocity of the pencil at time t = 0 seconds.

4.4 If your home were on the Moon, and a pencil were to drop 4 ft to the floor, its height function in feet would be y = 4 – 2.65t².

a) Set y = 0 and solve for t to determine that it requires approximately 1.2286 seconds for the pencil to hit the floor.

b) Determine the velocity the pencil strikes the floor.

c) Does a pencil on the Moon strike the floor faster, slower, or the same velocity as a pencil on Earth?

4.5 A person shoots an arrow vertically into the air at 200 ft/s. Neglecting air resistance, the height of the arrow after t seconds is given by the formula y = 6 + 200t – 16.1t² (approximately).

a) How high is the arrow when the bow is fired?

b) Find the instantaneous velocity when t = 0 s.

c) What is the instantaneous velocity of the arrow at time t = 6.21 s? What can you conclude about the maximum height the arrow reaches?

d) At what time t will the arrow hit the ground?

e) At what velocity does the arrow hit the ground?

f) Compare the velocity of the arrow striking the ground with the velocity of the arrow leaving the bow, and comment on the safety of firing an arrow into the sky.