Kinematics Notes

Acceleration

[Chapter 2 Objectives]

BHS -> Mr. Stanbrough -> Physics -> Mechanics -> Kinematics -> this page

Acceleration is the rate at which velocity changes. In other words, knowing the acceleration of an object tells you how fast the velocity of the object is changing.

Since velocity is the rate at which position changes, and acceleration is the rate at which velocity changes, acceleration is a "rate of a rate".

It is very important to notice that acceleration is defined in
terms of a change in *velocity - not speed*. This may seem
like a minor point, but it isn't. Since acceleration is the rate
velocity changes,

You are accelerating whenever you:

**speed up**- This is not difficult to grasp, since it is the common conception of acceleration.

**slow down**- People commonly call this deceleration, as if it were physically different from the "speeding up" case - but to a physicist, it's all the same - if your velocity is changing, you are accelerating.

**change direction**- This one can be tough to grasp. Since velocity is your speed and direction, your velocity changes if your direction changes (even if your speed stays the same). Since you are accelerating if your velocity is changing,**you are accelerating when you are changing direction - even if your speed stays the same**. Most people link acceleration with a change in speed (and increasing speed), so they think "OK, on a test I'm supposed to say that an object accelerates if it changes direction, but this is really some "fakey" kind of acceleration as opposed to the "real" speeding-up acceleration" Nope. This acceleration is just as real as the "speeding up" (and "slowing down") kind of acceleration.

Since acceleration is linked to changes in velocity - when your
velocity is changing, you are accelerating, students often get the
misconception that "acceleration is a change in velocity".
**NO!!** Acceleration is the rate at which the velocity
changes - there's an important difference!

To say that "acceleration is a change in velocity" is like saying
"velocity is a change in position" (This does sound silly, doesn't
it?). Velocity tells you how fast position is changing.
**Acceleration tells you how fast velocity is
changing.**

Acceleration is **not** a change in velocity!

Suppose a sprinter's velocity changes from 0 m/s to 10 m/s in 2 seconds at the start of a race. What is her acceleration?

Notice the odd units of acceleration - a distance (length) divided by 2 times. One time unit comes from the velocity in the numerator of the fraction, and the other comes from the denominator. Quantities like 4 mi/hr/sec are accelerations (4 mi/hr/sec means that the velocity changes by 4 mi/hr each second.).

A car starts from rest and accelerates at 2 m/s/s for 5 seconds. How fast will it be going?

The statement "car starts from rest" means that the car's starting velocity is 0 m/s. An acceleration of 2 m/s/s means that the car's velocity changes by 2 m/s each second. If the car's velocity starts at 0 m/s and changes by 2 m/s in the first second, it will be going 2 m/s after 1 second has elapsed. During the second second (?!), its velocity increases by 2 m/s - from 2 m/s to 4 m/s. Its velocity will be 6 m/s after 3 seconds, 8 m/s after 4 seconds, and 10 m/s after 5 seconds.

Alternatively, there is a more-algebraic approach if you prefer. If then , so . If the velocity starts at 0 m/s and changes by 10 m/s, it ends at 10 m/s.

Like velocity, accelerations in one-dimensional motions are positive if they act in the positive direction, and negative if they act in the negative direction. Determining the direction is a little more abstract for accelerations than velocities, however - at least until you get used to it.

There are 3 things to remember:

- The acceleration of an object is its change in velocity divided by the time.
- If an object is moving in the positive direction, its velocity is positive. If it is moving in the negative direction, its velocity is negative.
- Change in velocity is the ending velocity minus the starting velocity.

A car's velocity changes from +2 m/s to +10 m/s in 4 seconds. What is its acceleration?

The car's change in velocity = ending velocity - starting velocity
= 10 m/s - 2 m/s = 8 m/s. Its acceleration = its change in velocity
divided by the time taken = (8 m/s)/(4 s) = 2 m/s/s.

A car's velocity changes from +10 m/s to +2 m/s in 4 seconds. What is its acceleration?

The car's change in velocity = ending velocity - starting velocity
= 2 m/s - 10 m/s = -8 m/s. Its acceleration = its change in velocity
divided by the time taken = (-8 m/s)/(4 s) = -2 m/s/s.

A car's velocity changes from -2 m/s to -10 m/s in 4 seconds. What is its acceleration?

The car's change in velocity = ending velocity - starting velocity
= -10 m/s - (-2 m/s) = -10 m/s + 2 m/s = -8 m/s. Its acceleration =
its change in velocity divided by the time taken = (-8 m/s)/(4 s) =
-2 m/s/s.

A car's velocity changes from -10 m/s to -2 m/s in 4 seconds. What is its acceleration?

The car's change in velocity = ending velocity - starting velocity
= -2 m/s - (-10 m/s) = -2 m/s + 10 m/s = 8 m/s. Its acceleration =
its change in velocity divided by the time taken = (8 m/s)/(4 s) = 2
m/s/s.

From these examples, notice that you just follow the (simple)
rules - particularly remembering that the change in velocity is the
ending velocity minus the starting velocity - **NOT**
the big number minus the little number!

In the examples above, you saw that it is pretty straightforward
to calculate the acceleration of an object when its speed changes
using algebra. An object also accelerates when its direction changes,
but "normal" algebra is insufficient to calculate the acceleration in
this case - you need to use *vector algebra*, and we will not
bother with calculating these accelerations in this course.

Practice quiz on acceleration:

[Chapter 2 Objectives] BHS -> Mr. Stanbrough -> Physics -> Mechanics -> Kinematics -> this page

last update November 22, 2005 by JL Stanbrough