Acceleration and Speed Are Not the Same
In one-dimensional motion, velocity has a sign: it indicates the direction of motion. Speed does not have a sign; it measures how fast the object is moving. Acceleration describes how velocity changes, so its sign alone cannot tell you whether speed is increasing.
The key question is whether acceleration is acting in the same direction as the velocity or against it. If velocity and acceleration have the same sign, the magnitude of velocity increases and the object speeds up. If they have opposite signs, the magnitude of velocity decreases and the object slows down.
Velocity, Acceleration, and Speed
Let \(x(t)\) be an object's position at time \(t\). Its velocity is \(v(t)=x'(t)\), and its acceleration is \(a(t)=v'(t)\). The speed is the magnitude of velocity:
Thus, when \(v(t)>0\), the object moves in the positive direction and its speed equals \(v(t)\). When \(v(t)<0\), it moves in the negative direction and its speed equals \(-v(t)\). The minus sign makes speed positive even though velocity is negative.
At a time when \(v(t)\ne 0\), the rate of change of speed is
Because \(v(t)/|v(t)|\) is \(1\) for positive velocity and \(-1\) for negative velocity, this formula gives a simple sign test. A positive value of \(s'(t)\) means speeding up; a negative value means slowing down.
Why the Sign Test Works
The result follows directly from the definition of speed, considering the two possible signs of nonzero velocity.
If \(v(t)>0\), then \(s(t)=v(t)\). Differentiating gives \(s'(t)=v'(t)=a(t)\). In this case, speed increases when acceleration is positive and decreases when acceleration is negative.
If \(v(t)<0\), then \(s(t)=-v(t)\). Differentiating gives \(s'(t)=-v'(t)=-a(t)\). In this case, speed increases when acceleration is negative and decreases when acceleration is positive.
Combining the two cases gives the formula \(s'(t)=\frac{v(t)}{|v(t)|}a(t)\) whenever \(v(t)\ne0\). Equivalently, velocity and acceleration with the same sign give increasing speed; opposite signs give decreasing speed.
A Sign Guide
| Velocity \(v(t)\) | Acceleration \(a(t)\) | Effect on speed |
|---|---|---|
| Positive | Positive | Speeding up |
| Positive | Negative | Slowing down |
| Negative | Positive | Slowing down |
| Negative | Negative | Speeding up |
The signs must be considered together. Acceleration does not have to point in the direction of motion: when it points against the motion, it reduces speed.
Worked Examples
Worked Example: Positive Acceleration While Slowing Down
An object has velocity \(v(t)=2t-6\) metres per second, where \(t\) is measured in seconds. Determine whether it is speeding up or slowing down at \(t=1\), and verify the result using speed.
First evaluate the velocity:
Its acceleration is the derivative of velocity. For this linear velocity model, the difference quotient is
Therefore \(a(1)=2\ \text{m/s}^2\). The velocity is negative and the acceleration is positive, so their signs are opposite: the object is slowing down.
Near \(t=1\), the velocity remains negative, so speed is \(-v(t)=6-2t\). Its rate of change is \(-2\ \text{m/s}^2\). For a direct check, at \(t=1.1\) seconds, \(v(1.1)=2.2-6=-3.8\ \text{m/s}\), so the speed is \(3.8\ \text{m/s}\), down from \(4\ \text{m/s}\) at \(t=1\). This confirms that positive acceleration accompanies decreasing speed in this case.
Worked Example: Positive Acceleration While Speeding Up
An object's velocity is \(v(t)=1+3t\) metres per second. Decide what happens to its speed at \(t=1\).
The velocity at that time is
The acceleration follows from the difference quotient:
Thus \(a(1)=3\ \text{m/s}^2\). Both velocity and acceleration are positive, so the object is speeding up. Since velocity is positive nearby, speed equals \(v(t)\), and its rate of change is \(3\ \text{m/s}^2\).
Checking at \(t=1.1\) seconds gives \(v(1.1)=1+3(1.1)=4.3\ \text{m/s}\). The speed rises from \(4\ \text{m/s}\) to \(4.3\ \text{m/s}\), as predicted.
Worked Example: Negative Acceleration While Slowing Down
An object's velocity is \(v(t)=8-2t\) metres per second. Determine whether it is speeding up or slowing down at \(t=2\).
At \(t=2\), the velocity is
Its acceleration is
Therefore \(a(2)=-2\ \text{m/s}^2\). The velocity is positive but acceleration is negative, so the signs are opposite and the object is slowing down. In this time range speed equals \(8-2t\), whose rate of change is \(-2\ \text{m/s}^2\). At \(t=2.1\), the velocity is \(8-2(2.1)=3.8\ \text{m/s}\), so the speed has decreased from \(4\ \text{m/s}\) to \(3.8\ \text{m/s}\). This verifies the conclusion.
The Common Mistake
A frequent error is to read positive acceleration as “the object is speeding up.” For example, suppose \(v=-5\ \text{m/s}\) and \(a=2\ \text{m/s}^2\). The incorrect reasoning is: “Acceleration is positive, so speed is increasing.” That ignores the negative velocity.
Since the velocity is negative, speed is \(-v\) nearby. Its rate of change is therefore \(-a=-2\ \text{m/s}^2\), which is negative. The object is slowing down: its velocity is becoming less negative, so its magnitude is decreasing.
What Happens When Velocity Is Zero?
The sign test for the derivative of speed applies only when \(v(t)\ne0\). At a time when velocity is zero, the object is momentarily at rest, and the expression \(v(t)/|v(t)|\) is undefined. You must not label the object as speeding up or slowing down at that instant just by applying the four-row sign guide.
For example, if velocity crosses from negative to positive, speed can decrease to zero and then increase. If acceleration is positive at the crossing, velocity is increasing; that fact alone does not make the speed derivative exist at the crossing. The question of whether speed is changing should be answered using the motion immediately before and after that time, or by examining the speed function directly.
Why This Distinction Matters
In motion problems, acceleration often describes a change in velocity rather than a direct change in speed. A positive acceleration can slow an object moving in the negative direction; a negative acceleration can speed it up if it is already moving in that direction. Using both signs prevents confusing a change in direction with a change in how fast the object moves.
When interpreting a model, keep the units in view as well: velocity is measured in units of distance per time, while acceleration and the rate of change of speed are measured in units of distance per time squared. The sign describes direction of change; it does not change what quantity the unit measures.
Check Your Understanding
For each question, use velocity and acceleration together rather than relying on the acceleration sign alone.
- An object has \(v=7\ \text{m/s}\) and \(a=-3\ \text{m/s}^2\). Is it speeding up or slowing down?
- An object has \(v=-2\ \text{m/s}\) and \(a=-1\ \text{m/s}^2\). Is its speed increasing or decreasing?
- Explain why positive acceleration does not always imply increasing speed.
- When does the formula for the rate of change of speed in this tutorial apply?
- What extra care is needed when the velocity is zero?
