Velocity

If you have trouble with questions below involving functions, it may help to read more about functions.

1. A car is moving down a road. We can use highway markers to tell how far down the road the car is at any one time. Shown above is a table of where the car is at particular times.

   Note: This webpage uses math.js to parse your solutions to problems with numerical or algebraic answers. Math.js can handle units (which you should include when appropriate) but not long worded answers, so your answer should look like "30 miles" or "10 mi", not "5th mile marker". Only use units if the answer is a number, not a formula. Sorry if you have the correct answer, but the parser doesn't correctly recognize it!

   Where was the car 1.5 hours after the trip started? Check AnswerHintShow Answer

2. When was the car at mile marker 40? Check AnswerHintShow Answer

3. Did it take longer to travel from mile marker 0 to mile marker 60, or to return back? -Select your answer- To travel from mile marker 0 to mile marker 60 To return back from mile marker 60 to mile marker 0

4. Describe in your own words what might have happened during this trip.
   HintShow Answer

5. Let f be a function where we input some time into the function and get the position of the car at that time. That is, for any time t (in hours), f(t) is the position of the car at time t (in miles). What is f(2)? Check AnswerHintShow Answer

Velocity versus Speed

6. If the car is moving at a constant speed, we can compute that speed by taking the distance the car travels in a particular time divided by that amount of time. Suppose the car was moving at a constant speed between 1 hour and 1.5 hours into its trip. What is the distance between where the car was 1 hour into the trip and 1.5 hours into the trip? Check AnswerHintShow Answer

7. How much time elapsed? Check AnswerHintShow Answer

8. Divide your answers above to find the speed of the car. Check AnswerHintShow Answer Note that since the numbers on the table are in miles and hours, your answer is in miles per hour. Is this a reasonable speed for a car to be going?

9. Assume the car was moving at a constant speed between 3 hours and 3.2 hours into the trip. What is the distance between where the car was 3 hour into the trip and 3.2 hours into the trip? Check AnswerHintShow Answer

10. Divide your answer above by the time that elapsed to find the speed the car was going. Check AnswerHintShow Answer

    Since distance is always positive (or zero), speed is also always going to be positive (or zero). On the other hand, we aren't keeping track of which way the car is going. These two issues with speed can cancel each other out. Let's keep track of which way the car is going in the sign we're not using.

    Specifically, we will define two new quantities, called displacement and velocity. If the car is going in the direction of increasing mile markers (from mile marker 0 to mile marker 60) its displacement is just the distance it traveled. If the car is going in the direction of decreasing mile markers (from mile marker 60 to mile marker 0), then its displacement is just negative the distance it traveled.

11. What is the car's displacement between where it was 1 hour into the trip and 1.5 hours into the trip? Check AnswerHintShow Answer

12. What is the car's displacement between where it was 3 hours into the trip and 3.2 hours into the trip? Check AnswerHintShow Answer

13. If we let the variable s refer to the car's starting location, and e refer to the car's ending location, write down as nice a formula for the car's displacement as possible: Check AnswerHintShow Answer

    We then take the car's displacement and divide by the time elapsed to get its velocity. Note that, again, velocity is positive if the car is travelling in the direction of increasing mile markers and negative if the car is travelling in the direction of decreasing mile markers.

14. Assume the car was moving at a constant velocity between 1 hours and 1.5 hours into the trip. What is the car's velocity between when it was 1 hour into the trip and 1.5 hours into the trip? Check AnswerHintShow Answer

15. Assume the car was moving at a constant velocity between 3 hours and 3.2 hours into the trip. What is the car's velocity between when it was 3 hours into the trip and 3.2 hours into the trip? Check AnswerHintShow Answer

Computing Instantaneous Velocity

Our strategy for computing instantaneous velocity consists of two steps:

• Use average velocity to compute successively better and better approximations for instantaneous velocity.
• As we improve our approximations, they will get closer and closer to the true value, so we just need to figure out what value they're getting closer and closer to.

Caution: If we have a sequence of approximations, it's tempting to try to figure out what they're getting closer and closer to just by looking at their decimal digits. Eventually the decimal digits will "settle down" to the true decimal digits of the answer. This technique is far from ideal, though:

• We only find out some of the decimal digits of the answer, not a nice formula for the answer.
• Decimal digits can be deceptive as to when they're "settling down". They may appear to stop changing for a while, only to start changing again later on.

In many situations, we can use simple algebra to get a nice formula that is clearly the value our approximations are approaching.

20. Our strategy for computing an approximation to your instantaneous velocity at a particular time t will be to pick some small time interval containing time t and compute the average velocity of the car over that interval of time. Assume the variable h represents some positive, but small amount of time. Give a possible lower and upper bound for this interval in terms of t and h:
    Lower: Upper: Check AnswerHintShow Answer

    Don't forget: if your answer is a formula, don't include units. The poor parser can't handle it!

21. When you clicked "Go on" above, it unlocked a new feature in the car journey interactive under "Approximate Instantaneous Velocity". This will allow you to pick various interval formulas as above, and plot the computed approximation (the average velocity over the given interval) versus the actual instantaneous velocity. Which of the options for the intervals gives you the best approximation? -Select your answer- t,t+h t-h,t t-h,t+h

22. Where on the graph does using average velocity seem to give you particularly bad approximations for the instantaneous velocity?
    HintShow Answer

23. Which seems to be most important to getting a good approximation, choosing the right interval, or choosing a small value of h? -Select your answer- The right interval A small value of h

A Very Useful Formula

24. Let's go back to the situation where we use the function f to represent the position of the car as a function of time. That is, f(t) is the position of the car at time t. Write down a formula for the car's average velocity between time t and time t+h. Check AnswerHintShow AnswerStep-by-step

    1. Write down a formula for the position of the car at time t+h. Check AnswerHintShow Answer

    2. Write down a formula for the displacement between its position at time t and time t+h. Check AnswerHintShow Answer

    3. Write down a formula for the time elapsed between time t and t+h. Check AnswerHintShow Answer

    4. Write down a formula for the average velocity of the car between time t and a time h units after time t. Check AnswerHintShow Answer

25. We are interested in considering what happens to this formula above as h gets closer and closer to zero. What happens when h actually equals 0? HintShow Answer

An Example:
Your Initial Acceleration

26. Going back to your car journey above, if you accelerate leftwards right off the line, notice that your position as a function of time forms a parabola. This parabola is given by the formula f(t) = t²/5, where t is measured in minutes and f(t) is measured in feet. Plug this function into the formula above. Check AnswerHintShow AnswerStep-by-step

    1. Plug in t+h for t in the formula for f(t) to find f(t+h). Check AnswerHintShow Answer

    2. Subtract from your answer above the formula for f(t) to find f(t+h)-f(t). Check AnswerHintShow Answer

    3. If you haven't already, simplify your result. Check AnswerHintShow Answer

    4. Divide your answer above by h. Check AnswerHintShow Answer

27. Assuming h is not 0 (recall that we want h to be very small, but we don't want h to be zero), we can further simplify your answer above. Simplify as much as you can. Check AnswerHintShow Answer

28. So we've computed a nice formula for the car's average velocity between time t and t+h. Recall that as h gets smaller and smaller (closer to 0), this average velocity gets closer and closer to the true instantaneous velocity. As h gets smaller and smaller, what does your answer above get closer and closer to? Check AnswerHintShow Answer

29. We can now figure out the instantaneous velocity at whatever time we want. Plug in t = 3 to find the instantaneous velocity at 3 minutes. Check AnswerHintShow Answer

30. When is the instantaneous velocity 1 mile per minute? (i.e. how long does it take to go from 0 to 60 miles per hour? Note: your answer will be unrealistic.) Check AnswerHintShow Answer

Matching Speed

There's actually another way of figuring out your car's instantaneous velocity at a particular moment: drive a second car past such that:

• The second car matches the first car's position and velocity at the time in question.
• The second car is travelling at a constant velocity.

We then just measure the velocity of the second car. Since it has a constant velocity, its average velocity and instantaneous velocity are the same.

The car journey interactive above has a feature under "Matching Velocity" that allows you to visualize this second car.

Further Destinations

In this webpage, we focused on velocity, a measure of your change in position over time. But the techniques we used in this webpage can be extended to study almost anything that changes over time.

For instance, suppose you have a bank account that has continuously compounding interest. You might be intersted in figuring out how fast the amount of money you have is increasing at any one instant. Just like with our velocity situation:

• We can't use a simple change in money over change in time formula, since the rate at which the money in your account increases depends on how much money you have in your account. That is, the rate of change is itself changing.
• We have a formula for how much money you have in your account at any one time, just as we had a formula for your position at any one time. (see Wikipedia [Note])

Indeed, in many fields of study that involve numbers, those numbers change over time and the rate that they change changes over time, and we can use the techniques of this page to study how those numbers change. Can you think of some other examples of quantities that change over time that you might want to study?