Question

draw the velocity vs time graphs for an object whose motion produced the position vs time graphs shown below at left.
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1. for many graphs, both the slope of the line and the area between the line and the horizontal axis have physical meanings.

a. what does the slope of a position time graph tell you about the motion of an object?
b. looking at the velocity time graphs, determine the units for a square of area on the graph.
c. what does the area \under the velocity - time graph\ tell you about the motion of an object?

Explanation:

• a. The slope of a position - time graph represents the velocity of the object. Velocity is defined as the rate of change of position with respect to time ($v=\frac{\Delta x}{\Delta t}$), and the slope of a position - time graph ($m = \frac{y_2 - y_1}{x_2 - x_1}$) is calculated in the same way as the formula for velocity when $y$ represents position ($x$) and $x$ represents time ($t$).
• b. The units of velocity are meters per second ($m/s$) and the units of time are seconds ($s$). The area of a square on the velocity - time graph is calculated as $A = v\times t$. So the units of the area of a square on the graph are $(m/s)\times s=m$ (meters).
• c. The area under the velocity - time graph represents the displacement of the object. If the velocity is constant ($v$) over a time interval ($\Delta t$), the displacement ($\Delta x$) is given by $\Delta x=v\Delta t$, which is the area of the rectangle (a special case of the area under the graph) with height $v$ and base $\Delta t$. For a non - constant velocity, we can divide the time interval into small sub - intervals where the velocity is approximately constant over each sub - interval. Summing up the areas of these small rectangles (using integration in the limit of infinitely small sub - intervals) gives the total displacement.

Answer:

a. The slope of a position - time graph represents the velocity of the object.
b. The units for a square of area on the velocity - time graph are meters ($m$).
c. The area under the velocity - time graph represents the displacement of the object.