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Chapter 3Acceleration and Free Fall

If you know one of the changing variables and want to find another,

there is always an equation that relates those two:

v

x

t

a=

.

v

.

t

.

x = v

o

.

t + a

.

t

2

1

2

v

f

2

= v

o

2

+ 2a

.

x

The symmetry among the three variables is imperfect only because the

equation relating x and t includes the initial velocity.

There are two main difficulties encountered by students in applying

these equations:

•

The equations apply only to motion with constant acceleration. You

can’t apply them if the acceleration is changing.

•

Students are often unsure of which equation to use, or may cause

themselves unnecessary work by taking the longer path around the

triangle in the chart above. Organize your thoughts by listing the

variables you are given, the ones you want to find, and the ones you

aren’t given and don’t care about.

Example

Question: You are trying to pull an old lady out of the way of an

oncoming truck. You are able to give her an acceleration of 20 m/

s

2

. Starting from rest, how much time is required in order to move

her 2 m.

Solution: First we organize our thoughts:

Variables given:

.

x, a, v

o

Variables desired:

.

t

Irrelevant variables:v

f

Consulting the triangular chart above, the equation we need is

clearly

.

x=v

o

.

t+

1

2

a

.

t

2

, since it has the four variables of

interest and omits the irrelevant one. Eliminating the v

o

term and

solving for

.

t gives

.

t=2

.

x

a

=0.4 s.

Discussion questions

A Check that the units make sense in the three equations derived in this

section.

B. In chapter 1, I gave examples of correct and incorrect reasoning about

proportionality, using questions about the scaling of area and volume. Try to

translate the incorrect modes of reasoning shown there into mistakes about the

following question: If the acceleration of gravity on Mars is 1/3 that on Earth,

how many times longer does it take for a rock to drop the same distance on

Mars.