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Acceleration — Definition, Formula & Examples

Acceleration

The rate of change of velocity over time. For motion along the number line, acceleration is a scalar. For motion on a plane or through space, acceleration is a vector.

Key Formula

a=ΔvΔt=vf−vitf−tia = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{t_f - t_i}
Where:
  • aa = Average acceleration
  • vfv_f = Final velocity
  • viv_i = Initial velocity
  • tft_f = Final time
  • tit_i = Initial time
  • Δv\Delta v = Change in velocity
  • Δt\Delta t = Change in time

Worked Example

Problem: A car is traveling at 10 m/s. Over the next 4 seconds, it speeds up to 30 m/s. What is its average acceleration?
Step 1: Identify the initial and final velocities and the time interval.
vi=10 m/s,vf=30 m/s,Δt=4 sv_i = 10 \text{ m/s}, \quad v_f = 30 \text{ m/s}, \quad \Delta t = 4 \text{ s}
Step 2: Find the change in velocity.
Δv=vf−vi=30−10=20 m/s\Delta v = v_f - v_i = 30 - 10 = 20 \text{ m/s}
Step 3: Divide the change in velocity by the change in time.
a=ΔvΔt=204=5 m/s2a = \frac{\Delta v}{\Delta t} = \frac{20}{4} = 5 \text{ m/s}^2
Answer: The car's average acceleration is 5 m/s², meaning its velocity increases by 5 meters per second every second.

Another Example

Problem: A bicycle moving at 12 m/s applies its brakes and comes to a stop in 6 seconds. What is the average acceleration?
Step 1: Identify the known quantities. The bicycle starts at 12 m/s and ends at 0 m/s.
vi=12 m/s,vf=0 m/s,Δt=6 sv_i = 12 \text{ m/s}, \quad v_f = 0 \text{ m/s}, \quad \Delta t = 6 \text{ s}
Step 2: Calculate the change in velocity.
Δv=0−12=−12 m/s\Delta v = 0 - 12 = -12 \text{ m/s}
Step 3: Divide by the time interval to find acceleration.
a=−126=−2 m/s2a = \frac{-12}{6} = -2 \text{ m/s}^2
Answer: The average acceleration is −2 m/s². The negative sign indicates the bicycle is decelerating (slowing down).

Frequently Asked Questions

Can acceleration be negative?
Yes. Negative acceleration means the velocity is decreasing in the positive direction (or increasing in the negative direction). This is sometimes called deceleration. For example, a car slowing down from 20 m/s to 10 m/s has negative acceleration.
What is the difference between acceleration and velocity?
Velocity tells you how fast something is moving and in what direction. Acceleration tells you how quickly that velocity is changing. An object can have a large velocity but zero acceleration (constant speed in a straight line), or zero velocity but nonzero acceleration (a ball at the top of its arc, about to fall back down).

Velocity vs. Acceleration

Velocity measures the rate of change of position with respect to time, while acceleration measures the rate of change of velocity with respect to time. In calculus terms, if position is s(t)s(t), then velocity is the first derivative s′(t)s'(t) and acceleration is the second derivative s′′(t)s''(t). Velocity tells you where an object is heading; acceleration tells you how its motion is changing.

Why It Matters

Acceleration connects force to motion through Newton's second law, F=maF = ma, making it central to physics and engineering. In calculus, acceleration is the second derivative of the position function, so understanding it deepens your grasp of how derivatives describe real-world change. Any time you analyze a car braking, a rocket launching, or a ball in free fall, acceleration is the key quantity.

Common Mistakes

Mistake: Confusing acceleration with velocity, assuming that a fast-moving object must have high acceleration.
Correction: An object moving at constant velocity has zero acceleration, no matter how fast it travels. Acceleration only measures how velocity changes, not how large velocity is.
Mistake: Thinking that negative acceleration always means an object is slowing down.
Correction: Negative acceleration means velocity is changing in the negative direction. If an object is already moving in the negative direction, a negative acceleration actually makes it speed up. The sign depends on your chosen coordinate system.

Related Terms

  • Velocity — Acceleration is the rate of change of velocity
  • Scalar — Acceleration in one dimension is a scalar
  • Vector — Acceleration in two or three dimensions is a vector
  • Plane — Setting for two-dimensional acceleration vectors
  • Three Dimensions — Setting for three-dimensional acceleration vectors
  • Derivative — Acceleration is the second derivative of position
  • Speed — Magnitude of velocity, related but distinct from acceleration