Acceleration

In everyday conversation, to accelerate means to speed up. The accelerator in a car can in fact cause it to speed up. The greater the acceleration, the greater the change in velocity over a given time. The formal definition of acceleration is consistent with this understanding of acceleration, but acceleration also includes any change of velocity, not just speeding up.

To measure acceleration mathematically, the following equation should be used:

a= $LaTeX: \frac{\Delta v}{t}$

or

$LaTeX: a=\frac{v_f - v_i}{t}$

This shows that acceleration is equal to the change in velocity divided by time. To show a change in velocity you subtract the final velocity (v_f) - the initial velocity (v_i). The Δ is the Greek letter delta and stands for difference or change in.

Whenever you solve an equation, you must include the correct units. Acceleration is velocity in m/s divided by time in s, the SI units for acceleration are m/s², meters per second squared or meters per second per second, which literally means by how many meters per second the velocity changes every second.

Since the definition for acceleration is any change in velocity, this change could occur in the form of speeding up (increasing the velocity), slowing down (decreasing the velocity), or changing directions. When an object is decelerating, it will have a negative acceleration.

Acceleration Practice Problems

1. An object speeds up from a velocity of 240 meters/second to 560 meters/second in a time period of 10 seconds. What is the acceleration of the object?

Given:

Final velocity = v_f = 560 m/s

Initial velocity = v_i = 240 m/s

Time = 10 seconds

Solution:

a = (v_f - v_i ) / t =

(560 m/s - 240 m/s) / 10 sec =

320 m/s / 10 sec =

32 m/s²

a = 32 m/s²

Equation:

a = equation image indicator $LaTeX: \frac{v_f - v_i}{t}$

2. While traveling along a highway a driver slows from 24 m/sec to 15 m/sec in 12 seconds. What is the automobile's acceleration?

Given:

Final velocity = v_f = 15 m/s

Initial velocity = v_i = 24 m/s

Time = 12 seconds

Solution:

a = (v_f - v_i ) / t =

(15 m/s - 24 m/s) / 12 sec =

-9 m/s / 12 sec =

- 0.75 m/s²

a = - 0.75 m/s²

*Note since the car is slowing down (decelerating) there is a negative acceleration.*

Equation:

a= equation image indicator $LaTeX: \frac{v_f - v_i}{t}$

3. A cart rolling down an incline for 5.0 seconds has an acceleration of 4.0 m/sec². If the cart has a beginning speed of 2.0 m/sec, what is its final speed?

Given:

Initial velocity = v_i = 2.0 m/s

Time = 5.0 seconds

Acceleration = 4.0 m/s²

Solution:

a = (v_f - v_i ) / t =

4.0 m/s^{2 =} (v_f - 2.0 m/s) / 5.0 sec =

4.0 m/s² x 5.0 s= v_f - 2.0 m/s

20 m/s = v_f - 2.0 m/s

20 m/s + 2.0 m/s = v_f

v_f = 22 m/s

Equation:

a= equation image indicator $LaTeX: \frac{v_f - v_i}{t}$

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