Introduction
When an object moves along a line, its velocity tells us two things: how quickly its position is changing and which direction it is moving. Speed describes only the first of these. It measures the rate of travel without recording direction, so it cannot be negative.
The distinction is captured by one formula: instantaneous speed is the absolute value of instantaneous velocity. The formula is simple, but interpreting it correctly requires keeping track of what the sign and size of velocity mean.
Velocity and Speed on a Line
Suppose an object moves along a straight line, and let \(x(t)\) denote its position at time \(t\). The coordinate axis has a chosen positive direction. Instantaneous velocity is the derivative of position:
A positive velocity means position is increasing in the positive direction. A negative velocity means position is decreasing, so the object is moving in the negative direction. The sign depends on the direction chosen as positive.
Speed does not distinguish between those two directions. At a particular time, it is the size, or magnitude, of the velocity. For a real number \(a\), its absolute value \(|a|\) is \(a\) when \(a\geq 0\), and \(-a\) when \(a<0\). Thus:
Speed is measured in the same units as velocity, with the direction removed. If velocity is measured in metres per second, then speed is also measured in metres per second. For example, a velocity of \(-6\) metres per second corresponds to a speed of \(6\) metres per second, not a speed of \(-6\) metres per second.
Why Speed Is the Magnitude of Velocity
The formula follows from the meaning of velocity as a limit of average rates of change. Over a short time interval from \(t\) to \(t+h\), the signed change in position is \(x(t+h)-x(t)\), and the elapsed time is \(h\). The signed average velocity is therefore the change in position divided by the change in time.
To find the corresponding rate without direction, take the magnitude of that quotient. The absolute value of a quotient is the quotient of the absolute values, so the magnitude is the distance between the two positions divided by the length of the time interval. Taking the limit as the interval shrinks gives the instantaneous rate of travel.
If \(x\) is differentiable at \(t\), then the difference quotient tends to \(x'(t)\) as \(h\) tends to zero. The absolute value function is continuous, so the absolute values of the quotients tend to \(|x'(t)|\). This proves that the instantaneous rate of travel is the magnitude of the instantaneous velocity:
The sign of \(v(t)\) is needed to describe direction, but it disappears when the rate is expressed as speed. This is why changing the direction of motion changes the sign of velocity without requiring a negative speed.
Worked Examples from Position Functions
Worked Example: Positive velocity and a speed of 6 metres per second
An object's position, in metres, is \(x(t)=3t^2\), where \(t\) is measured in seconds. Find its velocity and speed at \(t=1\) second.
Differentiate the position function:
Substitute \(t=1\) second:
The velocity is positive, so the object is moving in the positive direction. Its speed is the magnitude of that velocity:
Check the derivative at this time using a difference quotient. Since \(x(1)=3\text{ m}\),
As \(h\to0\), this quotient tends to \(6\text{ m/s}\), agreeing with the calculated velocity. Its magnitude is \(6\text{ m/s}\), as stated.
Worked Example: Negative velocity with the same speed
Now let an object's position, in metres, be \(x(t)=-3t^2\), with time in seconds. Find its velocity and speed at \(t=1\) second.
Differentiate:
Evaluate the velocity at \(t=1\):
The negative sign means the object is moving in the negative direction. Speed is the magnitude:
To check, calculate the difference quotient. Here \(x(1)=-3\text{ m}\), so
As \(h\to0\), the quotient tends to \(-6\text{ m/s}\). This confirms the velocity, and its magnitude is \(6\text{ m/s}\). The two examples have opposite velocities but equal speeds.
Worked Example: Positive velocity and a different speed
An object's position, in metres, is \(x(t)=t^2\), where \(t\) is in seconds. Find its velocity and speed at \(t=2\) seconds.
Differentiate the position function:
Substitute \(t=2\) seconds:
The velocity is positive, and the speed is
Check the derivative value directly from the difference quotient. Since \(x(2)=4\text{ m}\),
As \(h\to0\), this tends to \(4\text{ m/s}\). Its magnitude is \(4\text{ m/s}\), matching the speed calculated above. This speed differs from the \(6\text{ m/s}\) in the first example.
Interpreting Velocity and Speed Together
A useful way to read a velocity value is to separate its sign from its magnitude. The sign gives direction relative to the chosen positive axis. The magnitude gives speed. For example, \(v(t)=-3\text{ m/s}\) means motion in the negative direction at \(3\text{ m/s}\).
| Velocity | Direction along the chosen axis | Speed |
|---|---|---|
| \(8\text{ m/s}\) | Positive direction | \(8\text{ m/s}\) |
| \(-8\text{ m/s}\) | Negative direction | \(8\text{ m/s}\) |
| \(2.5\text{ m/s}\) | Positive direction | \(2.5\text{ m/s}\) |
| \(-2.5\text{ m/s}\) | Negative direction | \(2.5\text{ m/s}\) |
Changing which direction is designated positive reverses the sign assigned to velocity, but it does not change the physical speed. The absolute value makes that distinction explicit.
A Common Mistake
A frequent error is to report a negative velocity as a negative speed. For instance, suppose \(v(t)=-5\text{ m/s}\). The incorrect calculation is to state that the speed is \(-5\text{ m/s}\), simply copying the velocity. That retains the direction sign, even though speed does not record direction.
When a velocity is given, take its absolute value to obtain speed. When a position function is given, first differentiate to find velocity and then take the absolute value. Taking the absolute value before differentiating is not the definition of speed; the defining quantity is \(|x'(t)|\).
Check Your Understanding
For each question, distinguish the signed velocity from the nonnegative speed.
- If an object's velocity is \(-9\text{ m/s}\), what is its speed, and what does the negative sign tell you?
- An object's position is \(x(t)=2t^2\) metres. Find its velocity and speed at \(t=3\) seconds.
- An object's position is \(x(t)=-4t^2\) metres. Find its velocity and speed at \(t=2\) seconds.
- Can two objects have different velocities but the same speed? Give a pair of velocities that demonstrates your answer.
- Why does the speed have the same units as velocity even though speed does not include direction?
