When calculating instantaneous velocity, we need to specify the explicit form of the position function
x (
t ). For the moment, let’s use polynomials
, because they are easily differentiated using the power rule of calculus:
The following example illustrates the use of
[link] .
Instantaneous velocity versus average velocity
The position of a particle is given by
.
Using
[link] and
[link] , find the instantaneous velocity at
s.
Calculate the average velocity between 1.0 s and 3.0 s.
Strategy
[link] give the instantaneous velocity of the particle as the derivative of the position function. Looking at the form of the position function given, we see that it is a polynomial in
t . Therefore, we can use
[link] , the power rule from calculus, to find the solution. We use
[link] to calculate the average velocity of the particle.
Solution
.
Substituting
t = 2.0 s into this equation gives
.
To determine the average velocity of the particle between 1.0 s and 3.0 s, we calculate the values of
x (1.0 s) and
x (3.0 s):
Then the average velocity is
Significance
In the limit that the time interval used to calculate
goes to zero, the value obtained for
converges to the value of
v.
Consider the motion of a particle in which the position is
.
What is the instantaneous velocity at
t = 0.25 s,
t = 0.50 s, and
t = 1.0 s?
What is the speed of the particle at these times?
Strategy
The instantaneous velocity is the derivative of the position function and the speed is the magnitude of the instantaneous velocity. We use
[link] and
[link] to solve for instantaneous velocity.
Solution
Significance
The velocity of the particle gives us direction information, indicating the particle is moving to the left (west) or right (east). The speed gives the magnitude of the velocity. By graphing the position, velocity, and speed as functions of time, we can understand these concepts visually
[link] . In (a), the graph shows the particle moving in the positive direction until
t = 0.5 s, when it reverses direction. The reversal of direction can also be seen in (b) at 0.5 s where the velocity is zero and then turns negative. At 1.0 s it is back at the origin where it started. The particle’s velocity at 1.0 s in (b) is negative, because it is traveling in the negative direction. But in (c), however, its speed is positive and remains positive throughout the travel time. We can also interpret velocity as the slope of the position-versus-time graph. The slope of
x (
t ) is decreasing toward zero, becoming zero at 0.5 s and increasingly negative thereafter. This analysis of comparing the graphs of position, velocity, and speed helps catch errors in calculations. The graphs must be consistent with each other and help interpret the calculations.
