3.4 Acceleration

Definition

Acceleration is the rate at which an object’s velocity changes with time.

Because velocity includes both magnitude and direction, an object accelerates whenever its velocity changes in magnitude, direction, or both.

An object therefore accelerates when it speeds up, slows down, or changes its direction of motion.

Since acceleration has both magnitude and direction, it is a vector quantity.

SI Unit

The SI unit of acceleration is the meter per second squared (m/s²).

An acceleration of 2 m/s², for example, means that the object’s velocity changes by 2 m/s each second.

Average Acceleration

The velocity of an object may change during a time interval. The average acceleration over that interval is defined as the change in velocity divided by the time taken.

a→avg=Δv→Δt\vec{a}_{avg}=\frac{\Delta\vec{v}}{\Delta t}

where

Δv→=v→f−v→i\Delta\vec{v}=\vec{v}_f-\vec{v}_i

so

a→avg=v→f−v→iΔt\vec{a}_{avg}=\frac{\vec{v}_f-\vec{v}_i}{\Delta t}

where:

a→avg\vec{a}_{avg}: average acceleration
v→i\vec{v}_i: initial velocity
v→f\vec{v}_f: final velocity
Δv→\Delta\vec{v}: change in velocity
Δt\Delta t: time interval

Example 1

A car is moving in a straight line with an initial velocity of 10 m/s. Its velocity increases to 25 m/s in 5 s.

Figure 3.4.1 – A car increasing its speed from 10 m/s to 25 m/s in 5 s.

The change in velocity is

Δv=25 m/s−10 m/s=15 m/s\Delta v=25\text{ m/s}-10\text{ m/s}=15\text{ m/s}

Therefore,

aavg=15 m/s5 sa_{avg}=\frac{15\text{ m/s}}{5\text{ s}}
aavg=3 m/s2a_{avg}=3\text{ m/s}^2

The car’s speed therefore increases by 3 m/s each second.

Direction of Acceleration

The direction of acceleration is the direction of the change in velocity.

For motion along a straight line, the sign of acceleration indicates its direction relative to the chosen positive direction.

If the positive direction is chosen to the right:

  • positive acceleration points to the right;
  • negative acceleration points to the left.

The sign of acceleration alone does not determine whether an object is speeding up or slowing down. The directions of both velocity and acceleration must be considered.

Figure 3.4.2 – The effect of acceleration depends on its direction relative to the velocity.

When velocity and acceleration point in the same direction, the object’s speed increases.

When velocity and acceleration point in opposite directions, the object’s speed decreases.

Therefore, negative acceleration does not necessarily mean slowing down.

Example 2

A car is moving to the right at 20 m/s. After 4 s, its velocity has decreased to 8 m/s.

Taking the rightward direction as positive,

vi=+20 m/sv_i=+20\text{ m/s}

vf=+8 m/sv_f=+8\text{ m/s}

The average acceleration is

aavg=8 m/s−20 m/s4 sa_{avg}=\frac{8\text{ m/s}-20\text{ m/s}}{4\text{ s}}

aavg=−3 m/s2a_{avg}=-3\text{ m/s}^2

The negative sign shows that the acceleration is directed to the left, opposite to the car’s velocity.

Since the velocity and acceleration are in opposite directions, the car is slowing down.

Example 3

An object can have a negative velocity while its acceleration is either positive or negative.

For example, suppose an object moves to the left and its velocity changes from −10 m/s to −20 m/s in 5 s.

aavg=−20 m/s−(−10 m/s)5 sa_{avg}=\frac{-20\text{ m/s}-(-10\text{ m/s})}{5\text{ s}}

aavg=−2 m/s2a_{avg}=-2\text{ m/s}^2

Both the velocity and acceleration are negative, so they point in the same direction. The object’s speed increases from 10 m/s to 20 m/s.

The object is therefore speeding up, even though its acceleration is negative.

Acceleration When Direction Changes

Acceleration can occur even when an object’s speed remains constant.

This happens when the direction of the velocity changes.

Figure 3.4.3 – An object can accelerate even at constant speed if the direction of its velocity changes.

Because velocity is a vector quantity, changing its direction changes the velocity. A change in velocity means that the object has acceleration.

This idea becomes particularly important in circular motion.

Instantaneous Acceleration

When acceleration changes during motion, instantaneous acceleration is the acceleration at a particular moment in time.

Average acceleration describes the change in velocity over a time interval, whereas instantaneous acceleration describes how rapidly velocity is changing at a particular instant.

For a very small time interval, the average acceleration approaches the instantaneous acceleration.

Instantaneous acceleration can be written as

a→=dv→dt\vec{a}=\frac{d\vec{v}}{dt}

This represents the rate of change of velocity at a particular instant.

Key Points

  • Acceleration is the rate at which velocity changes with time.
  • Acceleration is a vector quantity.
  • The SI unit of acceleration is the meter per second squared (m/s²).
  • Average acceleration is (a→avg=Δv→/Δt)(\vec{a}_{avg}=\Delta\vec{v}/\Delta t) .
  • Acceleration can result from a change in the magnitude or direction of velocity, or both.
  • The direction of acceleration is the direction of the change in velocity.
  • When velocity and acceleration have the same direction, the object speeds up.
  • When velocity and acceleration have opposite directions, the object slows down.
  • Negative acceleration does not necessarily mean slowing down.
  • An object can accelerate even when its speed remains constant if its direction of motion changes.
  • Instantaneous acceleration is the acceleration at a particular moment in time.

← 3.3 Velocity       

3.5 Position-Time Graphs →