Acceleration

Motion
definition

Also known as: speeding up, a

Grade 9-12

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The rate at which an object's velocity changes over time, measured in metres per second squared (m/s²). Acceleration is what forces produce, as described by Newton's second law (F = ma).

Definition

The rate at which an object's velocity changes over time, measured in metres per second squared (m/s²).

💡 Intuition

How quickly your speed (or direction) is changing. The 'push back' you feel when a car speeds up.

🎯 Core Idea

Acceleration is the derivative of velocity, the second derivative of position.

Example

Car goes from 0 to 60 mph in 10 seconds: a = 6 \text{ mph/s}

Formula

a = \frac{\Delta v}{\Delta t} (change in velocity divided by time)

Notation

\vec{a} is the acceleration vector in m/s², \Delta\vec{v} is the change in velocity in m/s, \Delta t is the time interval in seconds, and d\vec{v}/dt denotes the time derivative of velocity.

🌟 Why It Matters

Acceleration is what forces produce, as described by Newton's second law (F = ma). Understanding acceleration is essential for analysing car crashes, designing roller coasters, launching rockets, and predicting the motion of any object acted on by forces.

💭 Hint When Stuck

When solving an acceleration problem, first find the change in velocity \Delta v = v_f - v_i and the time interval \Delta t. Then divide: a = \Delta v / \Delta t. Pay attention to signs — a negative acceleration means the velocity is decreasing (or increasing in the negative direction).

Formal View

Average acceleration is \vec{a}_{\text{avg}} = \frac{\Delta\vec{v}}{\Delta t}, and instantaneous acceleration is \vec{a} = \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2}. Under constant acceleration, the kinematic equations are v = v_0 + at, x = x_0 + v_0 t + \frac{1}{2}at^2, and v^2 = v_0^2 + 2a\Delta x.

Compare With Similar Concepts

Speed vs Velocity

🚧 Common Stuck Point

Negative acceleration can mean slowing down OR speeding up backward.

⚠️ Common Mistakes

  • Thinking negative acceleration always means slowing down — if the object is moving in the negative direction, negative acceleration actually speeds it up.
  • Confusing acceleration with velocity — an object can have high velocity but zero acceleration (constant speed in a straight line), or zero velocity but nonzero acceleration (a ball at the top of its arc).
  • Forgetting that acceleration is a vector — in circular motion, there is centripetal acceleration even at constant speed because the direction changes.

Frequently Asked Questions

What is Acceleration in Physics?

The rate at which an object's velocity changes over time, measured in metres per second squared (m/s²).

What is the Acceleration formula?

a = \frac{\Delta v}{\Delta t} (change in velocity divided by time)

When do you use Acceleration?

When solving an acceleration problem, first find the change in velocity \Delta v = v_f - v_i and the time interval \Delta t. Then divide: a = \Delta v / \Delta t. Pay attention to signs — a negative acceleration means the velocity is decreasing (or increasing in the negative direction).

Prerequisites

velocity

How Acceleration Connects to Other Ideas

To understand acceleration, you should first be comfortable with velocity. Once you have a solid grasp of acceleration, you can move on to force and newtons second law.

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