Distinguish speed from velocity

A lift in a shaft

A lift moves in a shaft, with up as the positive direction. The graph shows its velocity v, in meters per second, after t seconds, and the line below it shows the height of the lift.

At t = 1 the velocity is 9, so the lift moves up at 9 meters per second.

At t = 7 the velocity is −9, so the lift moves down at 9 meters per second. The shading is green where the velocity is positive and red where it is negative.

Press Fold the graph up. The red part moves above the axis, and the folded graph shows the speed, which is how fast the lift moves in either direction.

At both times the folded graph has height 9, so the speed is 9 although the velocities are 9 and −9.

The distance from the axis

Drag the orange edge along the graph. The velocity is the height of the graph, so it is positive above the axis and negative below it.

Press Fold the graph up. The folded graph shows the speed, which is the distance of the graph from the axis, so it is never negative.

The speed is the size of the velocity without its sign, so it gives no direction.

Notation

1. First choose a positive direction. Motion in that direction has a positive velocity, and motion in the opposite direction has a negative velocity.

2. The speed is the absolute value of the velocity. It tells how fast the particle moves, but not which way.

   \[\text{speed} = \cy{|v(t)|}\]

3. Because of the absolute value, a velocity of −6 and a velocity of 6 both give a speed of 6.

   \[|{-6}| = 6\]

4. A speedometer shows speed, so it reads 30 miles per hour when a car drives out at 30 and when it drives back at 30. With out as positive, the velocity is +30 and then −30.

5. To find the speed at a time, find the velocity first. Then remove the sign.

Example 1: A cart

A cart has position \(s(t) = t^2 - 10t\) meters after \(t\) seconds. Find its velocity and its speed at \(t = 3\) and at \(t = 9\).

1. The velocity is the derivative of the position.

   \[v(t) = 2t - 10\]

2. At t = 3 the velocity is negative, so the cart moves in the negative direction at a speed of 4 meters per second.

   \[v(3) = \cr{-4}, \qquad |v(3)| = 4\]

3. At t = 9 the velocity is positive, so the cart moves in the positive direction at a speed of 8 meters per second.

   \[v(9) = \cg{8}, \qquad |v(9)| = 8\]

Example 2: Comparing two speeds

A particle has position \(s(t) = t^3 - 3t^2 - 9t\) meters after \(t\) seconds. Is it faster at \(t = 0\) or at \(t = 1\)?

1. Differentiate. Then substitute both times.

   \[\begin{gathered} v(t) = 3t^2 - 6t - 9 \\ v(0) = \cr{-9}, \quad v(1) = \cr{-12} \end{gathered}\]

2. A tempting wrong move: \(-12\) is less than \(-9\), so the particle is slower at \(t = 1\).

   At t = 1 the graph is farther below the axis, so after the fold it is higher there, at 12 instead of 9. The speed is the distance from the axis, not the value of v.

3. Compare the speeds, which are the absolute values of the velocities.

   \[|v(0)| = 9, \qquad |v(1)| = 12\]

4. The particle is faster at t = 1. It moves in the negative direction at both times.

Practice

A ball has height \(h(t) = 24t - 6t^2\) meters after \(t\) seconds, with up as the positive direction. Find its velocity and its speed at \(t = 3.5\). Say which way the ball moves.

Find the velocity from h(t). On the graph, the side of the axis gives the direction, and the distance from the axis is the speed.

Questions

These questions use a particle with velocity \(v(t) = (t - 1)(t - 5)\) meters per second. The graph shows \(v\), and the line below it shows the position of the particle.

1. At \(t = 3\) the graph is at height \(-4\). What is the particle doing?

   • It moves in the negative direction at a speed of 4 meters per second.
   • It moves at a speed of −4 meters per second.
   • It moves in the positive direction at a speed of 4 meters per second.
   • It is at rest.
   Answer

   It moves in the negative direction at a speed of 4 meters per second.

2. From \(t = 1\) to \(t = 5\) the graph is below the axis. When is the particle fastest in this interval?

   • At \(t = 3\), where the graph is farthest from the axis.
   • At \(t = 1\) and \(t = 5\), where \(v\) has its greatest value.
   • The speed is the same throughout the interval.
   Answer

   At \(t = 3\), where the graph is farthest from the axis.

3. A particle has position \(s(t) = 2t^2 - 12t\) meters. What is its speed at \(t = 1\)?

   • 8 meters per second
   • −8 meters per second
   • 10 meters per second
   • 4 meters per second
   Answer

   8 meters per second

Summary

To find the speed, you find the velocity and take its absolute value.

The folded graph shows why this works: the speed is the distance of the velocity graph from the axis, so it is never negative.

The sign of the velocity tells you which way a particle moves, and the speed tells you how fast.

This is one of five skills in topic 4.2 of Which & Why: AP Calculus.