# 0.2 Motion in one dimension  (Page 11/16)

 Page 11 / 16

To calculate the velocity of the taxi you need to calculate the gradient of the line at each second:

$\begin{array}{ccc}\hfill {v}_{1s}& =& \frac{\Delta x}{\Delta t}\hfill \\ & =& \frac{{x}_{f}-{x}_{i}}{{t}_{f}-{t}_{i}}\hfill \\ & =& \frac{5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}-0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}}{1,5\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}-0,5\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}}\hfill \\ & =& 5\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-1}\hfill \end{array}$
$\begin{array}{ccc}\hfill {v}_{2s}& =& \frac{\Delta x}{\Delta t}\hfill \\ & =& \frac{{x}_{f}-{x}_{i}}{{t}_{f}-{t}_{i}}\hfill \\ & =& \frac{15\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}-5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}}{2,5\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}-1,5\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}}\hfill \\ & =& 10\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-1}\hfill \end{array}$
$\begin{array}{ccc}\hfill {v}_{3s}& =& \frac{\Delta x}{\Delta t}\hfill \\ & =& \frac{{x}_{f}-{x}_{i}}{{t}_{f}-{t}_{i}}\hfill \\ & =& \frac{30\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}-15\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}}{3,5\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}-2,5\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}}\hfill \\ & =& 15\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-1}\hfill \end{array}$

From these velocities, we can draw the velocity-time graph which forms a straight line.

The acceleration is the gradient of the $v$ vs. $t$ graph and can be calculated as follows:

$\begin{array}{ccc}\hfill a& =& \frac{\Delta v}{\Delta t}\hfill \\ & =& \frac{{v}_{f}-{v}_{i}}{{t}_{f}-{t}_{i}}\hfill \\ & =& \frac{15\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-1}-5\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-1}}{3\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}-1\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}}\hfill \\ & =& 5\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-2}\hfill \end{array}$

The acceleration does not change during the motion (the gradient stays constant). This is motion at constant or uniform acceleration.

The graphs for this situation are shown in [link] . Graphs for motion with a constant acceleration (a) position vs. time (b) velocity vs. time (c) acceleration vs. time.

## Velocity from acceleration vs. time graphs

Just as we used velocity vs. time graphs to find displacement, we can use acceleration vs. time graphs to find the velocity of an object at a given moment in time. We simply calculate the area under the acceleration vs. time graph, at a given time. In the graph below, showing an object at a constant positive acceleration, the increase in velocity of the object after 2 seconds corresponds to the shaded portion.

$\begin{array}{ccc}\hfill v=\mathrm{area}\phantom{\rule{4pt}{0ex}}\mathrm{of}\phantom{\rule{4pt}{0ex}}\mathrm{rectangle}& =& a×\Delta t\hfill \\ & =& 5\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-2}×2\phantom{\rule{4pt}{0ex}}\mathrm{s}\hfill \\ & =& 10\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-1}\hfill \end{array}$

The velocity of the object at $t=2\mathrm{s}$ is therefore $10\phantom{\rule{2pt}{0ex}}\mathrm{m}·\mathrm{s}{}^{-1}$ . This corresponds with the values obtained in [link] .

## Summary of graphs

The relation between graphs of position, velocity and acceleration as functions of time is summarised in [link] .

Often you will be required to describe the motion of an object that is presented as a graph of either position, velocity or acceleration as functions of time. The description of the motion represented by a graph should include the following (where possible):

1. whether the object is moving in the positive or negative direction
2. whether the object is at rest, moving at constant velocity or moving at constant positive acceleration (speeding up) or constant negative acceleration (slowing down)

You will also often be required to draw graphs based on a description of the motion in words or from a diagram. Remember that these are just different methods of presenting the same information. If you keep in mind the general shapes of the graphs for the different types of motion, there should not be any difficulty with explaining what is happening.

## Aim:

To measure the position and time during motion and to use that data to plot a “Position vs. Time" graph.

## Apparatus:

Trolley, ticker tape apparatus, tape, graph paper, ruler, ramp

## Method:

1. Work with a friend. Copy the table below into your workbook.
2. Attach a length of tape to the trolley.
3. Run the other end of the tape through the ticker timer.
4. Start the ticker timer going and roll the trolley down the ramp.
5. Repeat steps 1 - 3.
6. On each piece of tape, measure the distance between successive dots. Note these distances in the table below.
7. Use the frequency of the ticker timer to work out the time intervals between successive dots. Note these times in the table below,
8. Work out the average values for distance and time.
9. Use the average distance and average time values to plot a graph of “Distance vs. Time" onto graph paper . Stick the graph paper into your workbook. (Remember that “A vs. B" always means “y vs. x").
10. Insert all axis labels and units onto your graph.
11. Draw the best straight line through your data points.

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