3.2 Instantaneous velocity and speed  (Page 2/7)

Finding velocity from a position-versus-time graph

Given the position-versus-time graph of [link] , find the velocity-versus-time graph.

Strategy

The graph contains three straight lines during three time intervals. We find the velocity during each time interval by taking the slope of the line using the grid.

Solution

Time interval 0 s to 0.5 s: $\stackrel{\text{–}}{v}=\frac{\text{Δ}x}{\text{Δ}t}=\frac{0.5\text{m}-0.0\text{m}}{0.5\text{s}-0.0\text{s}}=1.0\text{m/s}$

Time interval 0.5 s to 1.0 s: $\stackrel{\text{–}}{v}=\frac{\text{Δ}x}{\text{Δ}t}=\frac{0.0\text{m}-0.0\text{m}}{1.0\text{s}-0.5\text{s}}=0.0\text{m/s}$

Time interval 1.0 s to 2.0 s: $\stackrel{\text{–}}{v}=\frac{\text{Δ}x}{\text{Δ}t}=\frac{0.0\text{m}-0.5\text{m}}{2.0\text{s}-1.0\text{s}}=\mathrm{-0.5}\text{m/s}$

The graph of these values of velocity versus time is shown in [link] .

Significance

During the time interval between 0 s and 0.5 s, the object’s position is moving away from the origin and the position-versus-time curve has a positive slope. At any point along the curve during this time interval, we can find the instantaneous velocity by taking its slope, which is +1 m/s, as shown in [link] . In the subsequent time interval, between 0.5 s and 1.0 s, the position doesn’t change and we see the slope is zero. From 1.0 s to 2.0 s, the object is moving back toward the origin and the slope is −0.5 m/s. The object has reversed direction and has a negative velocity.

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Speed

In everyday language, most people use the terms speed and velocity interchangeably. In physics, however, they do not have the same meaning and are distinct concepts. One major difference is that speed has no direction; that is, speed is a scalar.

We can calculate the average speed    by finding the total distance traveled divided by the elapsed time:

Average speed is not necessarily the same as the magnitude of the average velocity, which is found by dividing the magnitude of the total displacement by the elapsed time. For example, if a trip starts and ends at the same location, the total displacement is zero, and therefore the average velocity is zero. The average speed, however, is not zero, because the total distance traveled is greater than zero. If we take a road trip of 300 km and need to be at our destination at a certain time, then we would be interested in our average speed.

However, we can calculate the instantaneous speed    from the magnitude of the instantaneous velocity:

If a particle is moving along the x -axis at +7.0 m/s and another particle is moving along the same axis at −7.0 m/s, they have different velocities, but both have the same speed of 7.0 m/s. Some typical speeds are shown in the following table.