How to Find Acceleration: 6 Methods, Formulas, and Examples
Acceleration is found by dividing the change in velocity by the change in time (a = Δv / Δt). The result is expressed in meters per second squared (m/s²). The 5 other methods use force and mass, distance and time, velocity and distance, graph slope, or circular motion data.
For a reliable reference, consult OpenStax section on acceleration.
What Is Acceleration and What Are Examples of Acceleration?
Acceleration is the rate at which the velocity of an object changes over time. It is a vector quantity, so it has a magnitude and a direction. The SI unit is meters per second squared (m/s²).
Acceleration occurs in 3 situations: speeding up, slowing down, and changing direction. Velocity changes in each case.
Moving objects show acceleration. Examples include cars, falling balls, braking bicycles, and orbiting satellites.
What Are the 6 Methods to Find Acceleration?
The 6 methods use velocity and time, force and mass, distance and time, velocity and distance, graph slope, or circular motion data. The table below lists the known quantities, formula, and conditions for each method.
Symbols in the table: v₀ is initial velocity (m/s), v is final velocity (m/s), t is time (s), F is net force (N), m is mass (kg), d is distance (m), and r is radius (m). All results are in m/s².
| Method | Known Quantities | Formula | Applies When |
|---|---|---|---|
| Velocity change | v₀, v, t | a = (v − v₀) / t | Velocity is measured at 2 moments |
| Newton's second law | F, m | a = F / m | Forces on the object are known |
| Distance and time | d, t, v₀ | a = 2(d − v₀t) / t² | Acceleration is constant |
| Velocity and distance | v₀, v, d | a = (v² − v₀²) / 2d | Time is unknown and acceleration is constant |
| Graph slope | Velocity-time graph | a = Δv / Δt along the line | A velocity-time graph is available |
| Centripetal acceleration | v, r | a = v² / r | Motion follows a circle |
How Do You Find Acceleration From Velocity and Time?
Acceleration is found by subtracting the initial velocity from the final velocity and dividing by the elapsed time (a = (v − v₀) / t). In the formula a = Δv / Δt, Δv is the change in velocity and Δt is the change in time.
- Identify the initial velocity (v₀) and the final velocity (v).
- Subtract the initial velocity from the final velocity.
- Identify the time interval (t) between the 2 velocities.
- Divide the velocity change by the time interval.
Example: A car speeds up from 10 m/s to 30 m/s in 5 s. The acceleration is (30 − 10) / 5 = 4 m/s².
How Do You Find Acceleration From Force and Mass?
Acceleration is found by dividing the net force by the mass (a = F / m). Newton's second law, published by Isaac Newton in 1687, links force, mass, and acceleration. According to NASA Glenn Research Center, force equals mass times acceleration for an object with constant mass.
Net force is the vector sum of all forces on an object. Forces include gravity, friction, tension, and thrust. Subtract friction from the applied force to find the net force, if friction acts on the object. The unit 1 newton equals 1 kg·m/s².
Example: A 12 kg crate is pushed with 80 N while friction applies 20 N. The net force is 80 − 20 = 60 N. The acceleration is 60 / 12 = 5 m/s².
How Do You Find Acceleration From Distance and Time?
Acceleration equals twice the distance divided by the time squared (a = 2d / t²), if the object starts from rest. The general formula is a = 2(d − v₀t) / t², if the starting velocity is not zero.
Galileo Galilei established this relationship in the early 1600s. His inclined plane experiments showed that distance covered from rest grows in proportion to the square of time, as published in Two New Sciences in 1638.
Example: A cart starts from rest and travels 50 m in 10 s. The acceleration is 2 × 50 / 10² = 1 m/s².
How Do You Find Acceleration From Velocity and Distance Without Time?
Acceleration is found by subtracting the squared initial velocity from the squared final velocity and dividing by twice the distance (a = (v² − v₀²) / 2d). This formula applies when time is unknown and acceleration is constant.
Example: A scooter increases speed from 5 m/s to 25 m/s over 100 m. The acceleration is (25² − 5²) / (2 × 100) = (625 − 25) / 200 = 3 m/s².
How Do You Find Acceleration From a Velocity-Time Graph?
Acceleration is found by calculating the slope of the line on a velocity-time graph. Slope equals rise divided by run, which is the change in velocity divided by the change in time.
Line shapes indicate the type of acceleration:
- A straight sloped line shows constant acceleration.
- A curved line shows changing acceleration.
- A horizontal line shows zero acceleration.
Example: A line rising from 0 m/s at 0 s to 20 m/s at 4 s has a slope of 20 / 4 = 5 m/s².
Graphs of position against time show acceleration through curvature. A curved line on a position-time graph indicates acceleration, and the second derivative of position gives its value.
How Do You Find Acceleration in Circular Motion?
Acceleration in circular motion is found by dividing the squared speed by the radius of the circle (a = v² / r). This is centripetal acceleration, and it points toward the center of the circle. It occurs at constant speed because the direction of velocity changes.
The equivalent form is a = ω²r, where ω is angular velocity in radians per second.
Example: A car travels at 10 m/s around a curve with a radius of 25 m. The acceleration is 10² / 25 = 4 m/s².
What Is the Difference Between Average and Instantaneous Acceleration?
Average acceleration is the total change in velocity divided by the total time interval. Instantaneous acceleration is the acceleration at a single moment. The table below compares the 2 types by definition, formula, and typical use.
| Type | Definition | Formula | Typical Use |
|---|---|---|---|
| Average acceleration | Change in velocity over a time interval | a = Δv / Δt | Trips and experiments with measured start and end velocities |
| Instantaneous acceleration | Acceleration at one moment in time | a = dv/dt = d²x/dt² | Motion described by a velocity or position function |
Example: If position is x(t) = t³ + 2t² (meters), then velocity is v(t) = 3t² + 4t and acceleration is a(t) = 6t + 4. At t = 2 s, the acceleration is 16 m/s².
How Do You Find Negative Acceleration and Deceleration?
Negative acceleration is found with the same formula (a = Δv / Δt), and the result carries a negative sign when the final velocity is lower than the initial velocity along the chosen positive direction.
Example: A cyclist slows from 12 m/s to 0 m/s in 4 s. The acceleration is (0 − 12) / 4 = −3 m/s².
Deceleration describes acceleration directed opposite to velocity. Negative acceleration does not always mean slowing down. An object moving in the negative direction and speeding up has negative acceleration.
How Do You Find Acceleration Due to Gravity?
Acceleration due to gravity is 9.80665 m/s² at the standard value on Earth, usually rounded to 9.8 m/s². The General Conference on Weights and Measures adopted this standard value in 1901.
Acceleration due to gravity is measured with the formula g = 2h / t², for an object dropped from rest. Example: A ball dropped from 1.5 m lands after 0.553 s. The value is 2 × 1.5 / 0.553² = 9.81 m/s².
Free-falling objects share the same acceleration regardless of mass, if air resistance is absent. In 1971, Apollo 15 commander David Scott dropped a hammer and a feather on the Moon, and both reached the surface at the same time.
The table below lists the surface gravitational acceleration of 3 bodies in m/s², rounded to 1 decimal place, based on NASA planetary fact sheets. The last column gives the distance fallen from rest in 1 s (d = ½gt²).
| Celestial Body | Surface Gravitational Acceleration (m/s²) | Distance Fallen in 1 s From Rest (m) |
|---|---|---|
| Earth | 9.8 | 4.9 |
| Moon | 1.6 | 0.8 |
| Mars | 3.7 | 1.85 |
How Do You Find Acceleration on an Inclined Plane?
Acceleration of an object sliding down a frictionless incline is found by multiplying the acceleration due to gravity by the sine of the incline angle (a = g sin θ). The symbol θ is the angle between the incline and the horizontal.
The formula with friction is a = g(sin θ − μ cos θ), where μ is the coefficient of kinetic friction.
Example: An object slides down a frictionless 30° incline. The acceleration is 9.8 × sin 30° = 9.8 × 0.5 = 4.9 m/s².
What Are the Units of Acceleration and How Are They Converted?
The SI unit of acceleration is meters per second squared (m/s²). The unit comes from dividing velocity (m/s) by time (s). The table below lists 5 units of acceleration and the value of each in m/s².
| Unit Name | Symbol | Value in m/s² |
|---|---|---|
| Meter per second squared | m/s² | 1 |
| Foot per second squared | ft/s² | 0.3048 |
| Standard gravity | g | 9.80665 |
| Kilometer per hour per second | km/h/s | 0.2778 |
| Centimeter per second squared (gal) | cm/s² | 0.01 |
Multiply the value by its m/s² equivalent to convert to m/s². Example: 32.2 ft/s² × 0.3048 = 9.81 m/s².
What Are the Common Mistakes When Finding Acceleration?
5 common mistakes cause incorrect acceleration results. Research by David Trowbridge and Lillian McDermott, published in the American Journal of Physics in 1981, found that introductory physics students commonly have difficulty distinguishing acceleration from velocity.
The mistakes include:
- Mixing units, such as km/h for velocity with seconds for time.
- Using speed instead of velocity, which removes direction.
- Dropping the sign, which hides deceleration and direction.
- Using one force instead of the net force in a = F / m.
- Applying constant-acceleration formulas to motion with changing acceleration.
Convert units before calculating. Example: 72 km/h equals 20 m/s.
How Do You Measure Acceleration Experimentally?
Acceleration is measured by recording distance or velocity against time with sensors or timers, then applying 1 of the 6 formulas. Sensors measure motion. Examples include accelerometers, photogates, and motion detectors.
Smartphones contain accelerometers that measure acceleration along 3 axes in m/s². A device at rest on Earth reads about 9.8 m/s², because accelerometers measure proper acceleration.
A ramp experiment follows 5 steps:
- Set up a cart on a ramp with a measured length.
- Release the cart from rest.
- Record the travel time with a stopwatch or photogate.
- Apply a = 2d / t² to the measured distance and time.
- Repeat 3 trials and average the results to reduce random error.
What Do People Also Ask About Finding Acceleration?
What is the formula for acceleration?
Acceleration is calculated with a = (v − v₀) / t, which divides the change in velocity by the elapsed time.
What is the unit of acceleration?
The SI unit of acceleration is meters per second squared (m/s²).
Can acceleration be negative?
Acceleration is negative when it points in the negative direction of the chosen axis.
How do you find acceleration without time?
Use a = (v² − v₀²) / 2d, which requires initial velocity, final velocity, and distance.
How do you find acceleration with mass and force?
Divide the net force in newtons by the mass in kilograms (a = F / m).
How do you find acceleration with distance and time?
Use a = 2d / t², if the object starts from rest.
How do you find acceleration from a graph?
Calculate the slope of the velocity-time graph by dividing the rise by the run.
What is the difference between speed and acceleration?
Speed measures how fast an object moves in m/s, while acceleration measures how quickly velocity changes in m/s².
What is the acceleration due to gravity on Earth?
Standard gravity on Earth is 9.80665 m/s², usually rounded to 9.8 m/s².
Which of the 6 Methods Finds Acceleration From the Known Quantities?
Acceleration is found by matching the known quantities to 1 of 6 formulas. Velocity and time give a = (v − v₀) / t. Force and mass give a = F / m. Distance and time give a = 2(d − v₀t) / t². Velocity and distance give a = (v² − v₀²) / 2d. A velocity-time graph gives acceleration as the slope. Speed and radius give a = v² / r. Each result is expressed in m/s² and carries a direction.
Title: What Is the Constant of Proportionality? Definition, 5 Methods, and Examples
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