Acceleration is the rate at which velocity changes. Because velocity is a vector, any change in either the speed or the direction of motion constitutes acceleration. Every time you brake in a car, ride a carousel, throw a ball, or fire a rocket, acceleration is at work. It is the physical quantity that links force to motion through Newton's second law (F = ma) — arguably the most important equation in classical mechanics.
Acceleration is the rate of change of velocity with respect to time. Formula: a = Δv/Δt. It is a vector quantity with both magnitude and direction, measured in metres per second squared (m/s²). Positive acceleration means velocity is increasing in the positive direction; negative acceleration (deceleration) means velocity is decreasing or directed opposite to the chosen positive direction.
The Acceleration Formula
The unit m/s² is intuitive: acceleration of 1 m/s² means velocity changes by 1 m/s every second. A car accelerating from rest at 3 m/s² will reach 3 m/s after 1 s, 6 m/s after 2 s, 9 m/s after 3 s, and so on.
Types of Acceleration
Uniform (constant) acceleration
Velocity changes by the same amount every second. Free fall near Earth's surface is the canonical example: in the absence of air resistance, every object falls with constant acceleration g = 9.8 m/s² downward. The SUVAT equations describe all constant-acceleration motion.
Non-uniform acceleration
Most real-world cases involve varying acceleration — a car in traffic, a rocket burning fuel (mass decreasing so acceleration increases for the same thrust). For non-uniform acceleration, instantaneous acceleration is:
Centripetal acceleration
For an object in circular motion with radius r at speed v:
This changes the direction of velocity without changing its magnitude. It keeps planets in orbit, cars on curved roads, and electrons in cyclotrons. By Newton's second law, centripetal force = mv²/r, directed toward centre.
Acceleration and Newton's Second Law
Acceleration is directly proportional to net force and inversely proportional to mass. Double the force → double the acceleration. Double the mass → halve the acceleration. A lorry accelerates more slowly than a car under the same engine force because it has greater mass.
| Scenario | Net Force | Acceleration |
|---|---|---|
| Free fall (no air resistance) | mg downward | 9.8 m/s² downward |
| Constant velocity | 0 N | 0 m/s² |
| Braking car | Friction backward | Negative (deceleration) |
| Circular orbit | Gravity toward centre | v²/r toward centre |
Acceleration Due to Gravity: g = 9.8 m/s²
Any object in free fall — regardless of mass — accelerates at g = 9.8 m/s². A feather and a hammer fall identically in vacuum, as Apollo 15 astronaut David Scott famously demonstrated on the Moon in 1971. This mass-independence was Galileo's great discovery and the seed of Einstein's general relativity.
g varies slightly: 9.832 m/s² at the poles, 9.780 m/s² at the equator, ~9.77 m/s² atop Everest. For most problems, g = 9.8 m/s² or g = 10 m/s² (approximate) is used.
Worked Examples
Example 1: Car acceleration
0 to 30 m/s in 10 seconds: a = (30 − 0) / 10 = 3 m/s²
Example 2: Braking (negative acceleration)
25 m/s to 0 in 5 seconds: a = (0 − 25) / 5 = −5 m/s²
Example 3: Newton's second law
1,200 kg car, 3,600 N net force: a = 3600 / 1200 = 3 m/s²
Example 4: Centripetal acceleration
Roundabout of radius 40 m at 15 m/s: a_c = 15² / 40 = 5.625 m/s² toward centre
SUVAT Equations for Constant Acceleration
Four equations link the five kinematic variables s (displacement), u (initial velocity), v (final velocity), a (acceleration), t (time):
Know any three variables → find the other two. These equations underpin all of projectile motion analysis and are among the most-used equations in introductory physics.
Frequently Asked Questions
The Acceleration Formula
Acceleration is the rate of change of velocity:
where v is final velocity, u is initial velocity, and t is time elapsed. The SI unit is metres per second squared (m/s²). Acceleration is a vector: it has both magnitude and direction. A negative acceleration (deceleration) means the object is slowing down in the positive direction — or speeding up in the negative direction.
Types of Acceleration
Uniform acceleration: constant rate of velocity change. Free fall (ignoring air resistance) has a = g = 9.8 m/s² downward — constant. SUVAT equations apply only for uniform acceleration.
Non-uniform acceleration: a changes with time. A car with increasing engine power accelerates faster over time. A falling object with air resistance decelerates as drag increases. Requires calculus: a = dv/dt.
Centripetal acceleration: changing direction at constant speed also constitutes acceleration. For circular motion at speed v and radius r: a_c = v²/r directed toward the centre. A car going around a bend at constant speed is accelerating — toward the centre of the curve.
Worked Example 1: Finding Acceleration
A car accelerates from 10 m/s to 30 m/s in 5.0 s. Find acceleration.
Worked Example 2: Deceleration
A train slows from 50 m/s to 20 m/s in 30 s. Find acceleration.
The negative sign confirms deceleration (opposing the direction of motion). The magnitude is 1.0 m/s².
Worked Example 3: Using Acceleration to Find Velocity
A ball rolls from rest and accelerates at 3.0 m/s² for 4.0 s. Find final velocity and distance covered.
Worked Example 4: g-Force
A fighter pilot pulls 6g in a tight turn. Find the acceleration and the apparent weight of a 80 kg pilot.
At 6g, blood pools in the lower body and the pilot may lose consciousness (G-LOC) without an anti-g suit.
Acceleration on Velocity-Time Graphs
On a v-t graph, acceleration = gradient of the line. Steeper gradient = larger acceleration. Horizontal line = zero acceleration (constant velocity). Negative gradient = deceleration. The area under the v-t graph = displacement (not distance, unless the object never reverses direction). For non-uniform acceleration, the instantaneous acceleration is the gradient of the tangent to the v-t curve at that moment.
Acceleration and Newton's Second Law
F = ma links force and acceleration directly. Net force causes acceleration — every acceleration in classical mechanics has a corresponding net force. No net force → no acceleration (Newton's first law). For a 1,000 kg car with 4,000 N engine force and 1,500 N resistance: F_net = 2,500 N → a = 2,500/1000 = 2.5 m/s².
Frequently Asked Questions
What is acceleration in physics?
Acceleration is the rate of change of velocity: a = Δv/Δt = (v−u)/t. It measures how quickly velocity changes, in both magnitude (speed) and direction. The SI unit is m/s². Acceleration is a vector — it has direction as well as magnitude. A negative value indicates the object is decelerating (if positive was the original direction) or accelerating in the negative direction. By Newton's second law, net force causes acceleration: a = F_net/m.
What is the difference between acceleration and deceleration?
Deceleration is not a separate concept — it's acceleration with a sign opposite to the direction of motion (the object is slowing down). In physics, we use signed acceleration: positive acceleration in the direction of motion means speeding up; negative acceleration (in the direction of motion) means slowing down. The word "deceleration" is colloquial. In equations, deceleration is simply a negative value of a when the positive direction is the direction of motion.
Can an object have zero velocity but non-zero acceleration?
Yes. A ball thrown upward has zero velocity at its highest point (momentarily stationary) but acceleration g = 9.8 m/s² downward throughout, including at the peak. If acceleration were also zero at the top, the ball would hover there indefinitely. The key insight: acceleration tells you how velocity is changing at that instant, not what the velocity is. Zero velocity doesn't mean zero acceleration; constant velocity (including zero) means zero acceleration.
What is centripetal acceleration?
Centripetal acceleration is the acceleration of an object moving in a circle at constant speed, directed toward the centre: a_c = v²/r. Even though speed is constant, the velocity vector's direction changes continuously — that change in direction is acceleration. For a car going around a 100 m radius bend at 20 m/s: a_c = 400/100 = 4 m/s² toward the centre of the bend, provided by friction. The centripetal force is F_c = mv²/r = m × a_c, directed inward.
How is acceleration shown on a velocity-time graph?
On a velocity-time (v-t) graph, acceleration equals the gradient (slope) of the line at any point. Steep positive gradient = large positive acceleration (rapid speeding up). Horizontal line = zero acceleration (constant velocity). Negative gradient = deceleration. The area under the v-t graph gives displacement. For uniform acceleration (straight line on v-t graph), acceleration is constant and all four SUVAT equations apply. For non-uniform acceleration (curved v-t graph), the instantaneous acceleration at any point is the gradient of the tangent there.
Measuring Acceleration Experimentally
Acceleration can be measured with: Light gates: two light beams a known distance apart; a card of known length breaks each beam; timing gives velocities at each gate; Δv/Δt gives acceleration. Video analysis: record motion at known frame rate; measure position each frame; displacement differences give velocity; velocity differences give acceleration. Accelerometer: a small proof mass attached to springs; acceleration displaces the mass, which is measured electrically. Modern smartphones contain MEMS accelerometers accurate to ~0.01 m/s². Ticker tape: a tape pulled through a timer at 50 Hz (50 dots per second); spacing between dots gives velocity; changing spacing reveals acceleration.
Acceleration in Special Relativity
Classical mechanics defines a = F/m with no upper limit on acceleration or final speed. Special relativity changes this: as v → c, the effective inertia increases without limit. A constant force produces decreasing acceleration as speed increases. The relativistic version: a = F/(γ³m) for acceleration parallel to velocity, where γ = 1/√(1−v²/c²). Even a constant force applied for infinite time never reaches c — the relativistic mass keeps increasing. This is why particle accelerators require enormous energies to push particles from 0.99c to 0.999c, even though the absolute speed gain is small.
What is acceleration in physics?
Acceleration is the rate of change of velocity with respect to time: a = Δv/Δt. It is a vector — it has magnitude and direction. Any change in speed or direction is acceleration. Its SI unit is m/s².
What is the formula for acceleration?
Average acceleration: a = Δv/Δt = (v_f − v_i)/t. Instantaneous: a = dv/dt. Centripetal: a_c = v²/r. All measured in m/s².
Can acceleration be negative?
Yes. Negative acceleration means the acceleration vector points opposite to the velocity vector, causing the object to slow down. It is not "less acceleration" — it is acceleration in a specific direction. Braking produces negative acceleration when forward is the positive direction.
What is the acceleration due to gravity?
Near Earth's surface, g ≈ 9.8 m/s² directed downward. All objects in free fall experience this acceleration regardless of mass. g varies slightly with altitude and latitude: ~9.78 m/s² at the equator, ~9.83 m/s² at the poles.
Can an object accelerate without changing speed?
Yes. Any change in direction is a change in velocity — and therefore an acceleration — even if speed is constant. An object in circular motion is continuously accelerating toward the centre, while its speed remains unchanged. This centripetal acceleration requires a centripetal force.
Is acceleration a vector or scalar?
Acceleration is a vector — it has both magnitude and direction. Its direction is the same as the net force on the object (from Newton's second law). Centripetal acceleration points toward the centre of the circle; gravitational acceleration points downward toward Earth's centre.
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