(a) Position:
x (
t ) versus time. (b) Velocity:
v (
t ) versus time. The slope of the position graph is the velocity. A rough comparison of the slopes of the tangent lines in (a) at 0.25 s, 0.5 s, and 1.0 s with the values for velocity at the corresponding times indicates they are the same values. (c) Speed:
versus time. Speed is always a positive number.
Check Your Understanding The position of an object as a function of time is
. (a) What is the velocity of the object as a function of time? (b) Is the velocity ever positive? (c) What are the velocity and speed at
t = 1.0 s?
(a) Taking the derivative of
x (
t ) gives
v (
t ) = −6
t m/s. (b) No, because time can never be negative. (c) The velocity is
v (1.0 s) = −6 m/s and the speed is
.
Instantaneous velocity is a continuous function of time and gives the velocity at any point in time during a particle’s motion. We can calculate the instantaneous velocity at a specific time by taking the derivative of the position function, which gives us the functional form of instantaneous velocity
v (
t ).
Instantaneous velocity is a vector and can be negative.
Instantaneous speed is found by taking the absolute value of instantaneous velocity, and it is always positive.
Average speed is total distance traveled divided by elapsed time.
The slope of a position-versus-time graph at a specific time gives instantaneous velocity at that time.
Conceptual questions
There is a distinction between average speed and the magnitude of average velocity. Give an example that illustrates the difference between these two quantities.
Average speed is the total distance traveled divided by the elapsed time. If you go for a walk, leaving and returning to your home, your average speed is a positive number. Since Average velocity = Displacement/Elapsed time, your average velocity is zero.
If you divide the total distance traveled on a car trip (as determined by the odometer) by the elapsed time of the trip, are you calculating average speed or magnitude of average velocity? Under what circumstances are these two quantities the same?
Average speed. They are the same if the car doesn’t reverse direction.
A woodchuck runs 20 m to the right in 5 s, then turns and runs 10 m to the left in 3 s. (a) What is the average velocity of the woodchuck? (b) What is its average speed?
An object has a position function
x (
t ) = 5
t m. (a) What is the velocity as a function of time? (b) Graph the position function and the velocity function.
A particle moves along the
x- axis according to
. (a) What is the instantaneous velocity at
t = 2 s and
t = 3 s? (b) What is the instantaneous speed at these times? (c) What is the average velocity between
t = 2 s and
t = 3 s?
Unreasonable results. A particle moves along the
x -axis according to
. At what time is the velocity of the particle equal to zero? Is this reasonable?
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