phys110 homework Average Velocity versus Instantaneous Velocity

PHYS-110 Written Homework #1

1. Average Velocity versus Instantaneous Velocity Consider an object moving along the x-axis. The textbook defines the average velocity during a

time interval Δt as ~vx,avg = �~x

�t . The textbook defines the instantaneous velocity at a specific

moment in time as ~vx = lim�t!0 �~x

�t . Let’s see if we can apply these equations to an example.

Consider an object moving along the x-axis with a position given by x = 3t2 (this corresponds to

an object undergoing constant acceleration). We can use this equation to calculate the object’s position (x) at any moment in time (t). Let’s make a table.

Time, t (seconds) Position, x (meters)

0 0

1 3

2 12

2.01

2.1

2.5 18.75

3

a) Use the equation x = 3t2 to fill in the missing position values in the table.

b) Consider the time interval from t = 2 s to t = 3 s. What is �x, �t, and v x,avg

for this interval?

Our v x,avg

value tells us the average velocity from t = 2 s to t = 3 s. But what if we want to know

the velocity exactly at t = 2 s? To find the instantaneous velocity, we use the same �x

�t expression but we look at what happens when we make �t smaller and smaller.

c) Consider the time interval from t = 2 s to t = 2.5 s. What is �x, �t, and v x,avg

for this interval?

d) Consider the time interval from t = 2 s to t = 2.1 s. What is �x, �t, and v x,avg

for this interval?

e) Consider the time interval from t = 2 s to t = 2.01 s. What is �x, �t, and v x,avg

for this interval?

f) Our answers for b), c), d), and e) seem to be getting closer and closer to a certain value. What value do they seem to be approaching?

You just took the limit of �x

�t as �t approaches zero! You started at t = 2 s (our moment of

interest) and you made �t smaller and smaller to see what value �x

�t approaches. In other words,

you found the instantaneous velocity at t = 2 seconds.

Too bad this process of finding the instantaneous velocity was so tedious. It would be nice if we didn’t have to do four different calculations and look for a trend. It would be great if we could somehow find the instantaneous velocity in one step. That’s what Calculus allows us to do! In Calculus, the limit process you just went through is called “taking the derivative of x with

respect to t” and is written as dx

dt

. It turns out there is an easy rule to figure out what your answer

is going to be:

If x = Ct2 where C is any number, then lim�t!0

�x

�t =

dx

dt

= 2Ct .

g) Our position equation is x = 3t2. What is the value of C for our position equation?

h) What value do you get for 2Ct if you use your value for C from part g and plug in 2 for t? Is this the same as (or very close to) your answer from part f?

In our motion diagrams, �t between points is usually very small which is why we usually just write ~v instead of ~vavg. The equation we used to find the limit in this problem (2Ct) is an example of the general equation shown at the end of Section 2.4 in our textbook.

2. A Ball on a Ramp Consider a ball that rolls up and back down a short ramp. a) Sketch a motion diagram for the ball’s entire up and back down trip. b) Add velocity vectors to your motion diagram. c) Explain in words and show with a sketch how you determine the acceleration vector. d) Add an acceleration vector for while the ball is moving up the ramp and a second acceleration

vector for when the ball is coming down the ramp. e) For any problem, how should the directions of ~v and ~a compare when an object is speeding

up? How should they compare when an object is slowing down? Are the vectors in your motion diagram consistent with these ideas?

f) Sketch a position vs. time graph for the ball (position vs. time means that position is on the vertical axis and time is on the horizontal axis).

g) Sketch a velocity vs. time graph for the ball. *There are no numbers in this problem, but your graphs should be accurate with regard to: - Whether the graph starts at positive values or negative values

- Whether the graph is a straight line or is curved - Whether the graph ever crosses the time axis h) On BOTH GRAPHS indicate the moment when the ball changes direction. i) When the ball changes direction, is the slope of your position graph positive, negative, or

zero? What does this tell you about the velocity at that moment? j) When the ball changes direction, is the slope of your velocity graph positive, negative, or

zero? What does this tell you about the acceleration at that moment?

3. Translating Between Graphs Consider the position vs. time graph below.

a) Suppose x = 0 is the location of your car. Write a short “story” of the motion being depicted in this graph.

b) Sketch the corresponding velocity vs. time graph with accurate numerical values.