Exercises I.B.

In problems 1 through 12 assume that $s(t)$ is the position of an object moving on a coordinate line.

a) Find the average velocity of the object for the time intervals [1,2], [1,1.1], [0,1], and [.9,1]. Sketch a transformation figure to illustrate the data used for these calculations.

b) Find the instantaneous velocity at times $t = 1$, $t = -1$, and $t=a$.

1.         $s(t) = 5t^2 - 4t + 1$.

2.         $ s(t) = 3t^2 - 2t.$

3.         $s(t) = 7 -t^2.$

4.         $s(t) = 2 - t  - 3t^2$.

5.         $s(t) = 3t - 2$.

6.         $s(t) = 3 - 4t$.

8.         $s(t) = 5$.

9.         $s(t) = 2t^3 - t$ .

10.     $s(t) = 2t^3 - t^2$.

11.     $s(t) = t^4.$

12.     $ s(t) = 1/(t + 5)$  with $t \ne -5$.

13.     $s(t) = 1/(t - 3)$  with $t \ne 3$.

14.     Suppose a spring with a weight attached to it is moving up and down so that its position above the ground at time t is given by s(t) = 2 sin(t) + 3 meters. Estimate the instantaneous velocity of the weight at time t = 0 using average velocities for the time intervals between 0 and x where x =.2, .1, .05, -.2, -.1, and -.05 seconds. Discuss briefly what you think the instantaneous velocity might be at time t = 0.

15.     Sketch transformation figures for s(t) = t 2 illustrating the average velocity of an object with its position on the target line determined by s where t = 2 and x = 1.9, 1.99, 1.999, 2.1, 2.01, and 2.001. Each of these should use a different view to allow the two times and positions to be visible easily.

16.     For each of the following position functions use your calculator to estimate the instantaneous velocity at time t=0 using an average velocity for the time interval [0,.001].

a.                  s(t) = 2 t.

b.                  s(t) = 3 t

c.                  s(t) = (.5) t

d.                  s(t) = (1/3) t.

The smooth transfer. Sometimes we want objects moving at different speeds to be at the same point travelling at the same instantaneous velocity at the same time so a transfer can be made smoothly between the two objects.

17.     Suppose two objects are moving on a coordinate line. At time t the first object is at position s(t) = t 2 while the second object is moving at a constant velocity and at time 3 has its position at 5. Find any possible velocities for the second object so that at some time the two objects might be at the same position and traveling at the same velocity. Discuss briefly the strategy you followed in arriving at your conclusions.

18.     Suppose two objects are moving on a coordinate line. At time t the first object is at position s(t) = t 2 +2 while the second object is moving at a constant velocity and at time 2 has its position at 10. Find any possible velocities for the second object so that at some time the two objects might be at the same position and traveling at the same velocity. Discuss briefly the strategy you followed in arriving at your conclusions.

19.     An object moves on a coordinate line with its position at time t seconds determined as s(t) feet where s(t) = t 2 - 4.

a) Find the instantaneous velocity and speed of the object when t is 3 and -3 seconds.

b) Find the instantaneous velocity and speed at those times t when s(t) = 0.

c) Find the time t (if any) when the instantaneous velocity at t will be the same as the average velocity for the interval [ 1,4 ].

20.     An anvil dropped from the top of a mesa has its position s(t) feet above ground level at time t seconds, where s(t) = -16t 2 + 144 feet. a) How high is the anvil after 1 second?

b) What is the (instantaneous) velocity at 1 second?

c) When will the anvil be at ground level?

d) What is the velocity when the anvil reaches ground level?

e) What is the average velocity of the anvil for the time interval it fell to ground level?

f) Find a time when the instantaneous velocity of the anvil was equal to its average velocity for the time interval it fell to ground level.

21.     Stopped for an interval, stopped for an instant. Write a short story about something that is moving and comes to a stop, remains stopped for a period of time and then starts to move again. In the same story describe a situation where something slows down and stops for only an instant before starting up again. Discuss what the situations means in terms of the average and instantaneous velocities during the intervals when the stopping happened.

22.     Historical projects: Two people who were very interested in understanding motion with a mathematical model were Nicole Oresme (1320-1382) and Galileo Galilei (1564-1642). Investigate the writings of one of these two scientists and write a short paper describing at least one problem of motion which they studied; what was  their approach to the study; and how they stated their results. 




Time Arrived


Distance from S.F.


Santa Cruz


9:40 am


75 miles




11:25 am


122 miles


San Luis Obispo


3:20 pm


230 miles


Santa Barbara


6:10 pm


336 miles


Los Angeles


9:25 pm


430 miles

23.     On a trip from San Francisco to Los Angeles on U.S. Route 1, the road conditions for driving vary dramatically with steep hills and winding curves in some sections followed by open straight sections that sometimes have no other entering roads for miles. We left San Francisco at about 7:45 am and  stopped for about 45 minutes at a beach between Carmel and Big Sur for lunch. Later in the day we turned off the road for about a half hour later just north of Santa Barbara to look at the ocean and the surf pounding on the coast. The table in Figure *** shows some of the cities along the way, the distance travelled from San Francisco to these cities, and the time we reached these cities on the trip. Draw a  transformation figure and a graph that might fit this information. Find the average speed for each of the segments of the trip between these cities. Based on this information, in which segments of the trip do you think the road was most hazardous? Discuss how you dealt with the delays for lunch and sightseeing.  Are there things we may have forgotten to tell you about the trip that would change your conclusions on where the road was dangerous? Discuss how these might change your conclusions.

24.     I forgot to tell you in problem 22 that we took a wrong turn just south of  Santa Barbara that took us 15 minutes to figure out and correct and took us 8 miles out of the way. How does this effect the average speed you calculated?






9:00 am


55 mph


9:45 am


60 mph


10:30 am


70 mph


11:15 am


60 mph


12:00 pm


65 mph


12:45 pm


55 mph

 25.     The odometer and the gas gauge on my car have not worked for years. (I bought the car used and they didn't work when I bought the car.) This isn't a problem for me when I drive around town. I just make it a point to fill the gas tank  on Mondays,  Thursdays, and Saturdays and I never need more than 8 gallons. The car has a 13.5 gallon gas tank, so this works fine. On long trips though I need another technique. What I do is figure I get about 20 miles a gallon on the highway, so I try to fill my tank every 200 miles to be on the safe side. To estimate how many miles I've driven I watch my speedometer (which still works fairly accurately) as I drive along the highway and try to travel at the same speed for 45 minute periods. On long trips I do vary my speed to help break up the monotony. Here is a table of my speeds that I recorded on a recent trip across the country. Before I started the table I estimated I had travelled 40 miles starting with a full tank of gas. Based on this table, when do you think I filled my gas tank?

26.     In many sports, speed is essential to winning competitions or enjoying the activity. Write a brief essay on one of the following sports and the way that speed is involved and what speeds are actually involved: waterskiing, snowboarding, swimming, jogging, diving, marathon running, football, tennis, soccer, softball, golf, baseball (running bases), bowling.

27.     Quarter Horse Race. 

28.     Coming home from work- with transformation figure. Transfer info to table and graph.

29.     The Bicycle Ride. The following graph shows the distance of a bicyclist from the starting point of a 50 mile excursion tour of the Avenue of the Giants from start to finish. Based on this graph complete the following table and sketch a transformation figure showing some locations at what you consider the important points in time. Write a short story to go along with this graph that explains what was happening at some of these times.



Explore reversal to accumulate estimates of change based on velocities, problems that have other real situations of interest for estimating distance travelled or time elapsed.

30.     Estimate distance traveled based on information about speed at various times from tables.

31.     Estimate position function with TF from Tf of velocity.

32.     Choose graphs of velocity functions based on position functions.

33.     Sketch graph of position function based on graph of velocity.

34.     A problem about speed and growth rates-- maybe corn or beans, redwoods, children.

35.     Project: Investigate how rockets take off and reach "escape velocity."

36.     using lengths to keep track of other variables.

Martin Flashman, Feb. 1, 2014. Created with GeoGebra