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Acceleration

When an object's velocity vector is changing, the object is accelerating. Examples include a car speeding up ("accelerating" in common parlance), slowing down or braking ("decelerating", but physicists prefer to say "negatively accelerating"), and turning. In three dimensions, we define acceleration by

(3a)

(3b)

The numerator for equation (3a) gives the change in the velocity vector, so there is an acceleration if either the magnitude or the direction of the velocity vector change. We will talk more about this in Chapter 6. In one dimension the definition of acceleration is
 
  (4a)
 
   (4b)

The units for acceleration are [(m/s)/s = m/s · 1/s = m/s2].

 

Example

Take north to be positive. A car is traveling south and speeding up. What is the sign of the acceleration?

Solution

Since the velocity vector points south and the car is speeding up, the acceleration vector must point south. With this sign convention, acceleration is negative.

 

Example

Take north to be positive. A car traveling south speeds up from 10 m/s to 15 m/s in 10 s. What is its acceleration?

Solution

We write

 

 

This confirms our thinking in Example 1a.
 

 

Example

Take north to be positive. A car is traveling north and slowing for a red light. What is the sign of the acceleration?

Solution

The velocity vector points north. Since this vector is shrinking, the acceleration vector must point south. Thus the acceleration is negative.
 

 
 

Example

What is the acceleration for the car in Example 2a slowing from 10 m/s to 8 m/s in 1 s?
 

Solution

We write

 

 

 

 

Example

An Oldsmobile takes a certain amount of time to accelerate from 0 to 60 mph. A Porsche takes less time by a factor of 3 to accelerate from 0 to 60 mph. How does the Porsche acceleration compare with that of the Oldsmobile?

Solution

We look at equation (4)

 

Since Δv is constant, if Δt is smaller by a factor of 3, then a is larger by a factor of 3.
 

 





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