Wednesday, February 1, 2017

Displacement and Acceleration

If the relationship between displacement, acceleration and elapsed time is extended to the most general case, the following equations need to be combined.

1.)
df = di + vt

where df is the final displacement from the fixed point, di is the initial displacement from the fixed point, v is rate of change in position, (average velocity), and t is elapsed time.

2.)
Vf = Vi + at

where Vf is final velocity, Vi is initial velocity, a is the rate of acceleration, and t is the elapsed time.

3.)

Vave = (V + Vi) / 2

where Vf is final velocity, Vi is initial velocity, and Vave is the average velocity.

To begin, start with the general displacement equation:

df = di + vt

Understanding that v is average velocity, the equation becomes:


df = di + ((V + Vi) / 2) • t

And since Vf can be found in relationship to acceleration, the following emerges:


df = di + (((Vi + at) + Vi) / 2) • t


Solving the problem in steps generally is more easily understood. To find displacement when acceleration is present, do the following.

Step 1.) Find the final velocity: Vf = Vi + at

Step 2.) Find the average velocity: Vave = (V + Vi) / 2

Step 3.) Find the displacement: df = di + vt

EXAMPLE

A model car is 3 meters from the starting line on a model car race track and it is moving at 2 m/s. It accelerates at a rate of 5 (m/s)/s for 4 seconds. How far does it end up from the starting line?

Step 0.) Collect the data!

di = 3 m
vi = 2 m/s
a = 5 (m/s)/s
t = 4

Step 1.) Find final velocity.

Vf = vi + at
vf = 2 + 5•4
vf = 2 + 20
vf = 22

Step 2.) Find average velocity.

v(ave) = (vf + vi)/2
v(ave) = (22 + 2)/2
v(ave) = 24/2
v(ave) = 12 m/s

Step 3.) Find the total displacement.

df = di + vt

(remember v is average velocity and t is the same elapsed time as for the acceleration)

df = 3 + 12•4
df = 3 + 48
df = 51 meters 

So, the model car ends up 51 meters from the starting line.

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