Position and velocity.

Step 1 of 6

Position with a sign

A measuring tape in metres runs along the track, with its zero at the blue start line. The cart’s position, $x$⁠, is the number on the tape beneath it. The motion sensor stands at the near end of the track, and positive is away from it. So a position on the sensor’s side is a negative number.

In the scene the cart crosses the start line heading for the sensor, rolls up to the end stop and stops there. The end stop is 0.6 m from the start line. The position graph beside the scene plots $x$⁠ against time.

Step 2 of 6

Without a push

Now the fan is off and the cart is standing on the start line. Its position graph is a flat line at zero: time passes and the position does not change.

In a moment the cart will cross the start line at 0.3 m/s, away from the sensor, and the fan will stay off. There is no friction in this model. Predict what the position graph will look like, and only then watch.

Step 3 of 6

Velocity is the slope

A cart moving at a constant velocity covers the same distance every second, so its position changes by the same step every second: the position graph is a straight line. This is uniform motion, motion at a constant velocity.

In the linear functions Unit you saw that the slope of a line is its rise over its run. On a position–time graph the rise is the change in position, $\Delta x$⁠, and the run is the time that passed, $\Delta t$⁠. Their ratio is the velocity:

$v = \frac{\Delta x}{\Delta t}$$v =$$\frac{\Delta x}{\Delta t}$⁠

Its unit is metres per second, m/s. And just as $y = mx + n$⁠, the position in uniform motion is

$x = x_0 + vt$⁠

where $x_0$⁠ is the position when $t = 0$⁠, the intercept on the vertical axis, and $v$⁠ is the slope. In the scene $x_0 = 0$⁠, because the clock starts as the cart crosses the start line.

Step 4 of 6

Negative velocity

When the cart travels towards the sensor, its position gets smaller with time and the position graph slopes down. The slope is negative, so the velocity is negative too. The sign of the velocity is the direction; the number without its sign is the speed, how fast.

Starting velocity is hidden now. Look at the position graph: where is the cart after the four seconds, and what velocity does that mean?

Step 5 of 6

The equation of uniform motion

$x = x_0 + vt$⁠ describes any motion at a constant velocity, not just a cart on a track. The starting position is wherever the object is when you start timing, and it need not be zero. The velocity is negative when the object moves in the negative direction.

Try it: move Starting velocity between 0.6 m/s and −0.4 m/s, and watch the position graph change its slope. The steeper the line, the faster the cart.

Back to the experiment ↑

Step 6 of 6

Summary

• Position $x$⁠ is measured from a zero point, with a sign that says which side of it the object is on.
• In uniform motion the position graph is a straight line, $x = x_0 + vt$⁠, and its slope is the velocity, $v = \dfrac{\Delta x}{\Delta t}$$v =$$\dfrac{\Delta x}{\Delta t}$⁠.
• A negative velocity means motion in the negative direction; the speed is the number without its sign.
• In the next Lesson the fan comes on, and the velocity itself starts to change.