Chapter 3

3.1 Acceleration

How do you know when velocity is changing? What do you experience?

Particle-models can represent velocityEvenly spaced dots = constant velocityDots spreading further apart = speeding upDots moving closer together = slowing down

Changing Velocity

Motion Diagrams

On a motion diagram, velocity is represented using average acceleration vectors.

To find average acceleration, subtract two consecutive velocity vectors. Then divide by time interval.

The difference between two velocity vectors represents

ΔV

Positive and Negative AccelerationPositive Acceleration Negative Acceleration

Velocity vectors and acceleration vectors point in the same direction

Object speeds up in the positive direction

Slowing down in the negative direction

Velocity vectors and acceleration vectors point in opposite direction

Object slows down in the positive direction

Object speeds up in the negative direction

This will all be easier to see on an Velocity – Time Graph!

Be Careful!Sign of acceleration DOES NOT indicate

speeding up or slowing down!

Velocity – Time GraphsWhat can we determine about the motion of an object by looking at a

Velocity – Time graph?

Velocity – Time Graphs Slope of line represents acceleration. Average acceleration is defined as the

change in velocity during some measurable time interval divided by that time interval.

Instantaneous acceleration is the change in velocity at an instant of time.Found by drawing a line tangent to the time you

are interested in on a velocity-time graph.We generally will not solve for instantaneous

acceleration in this class. (Calculus!)

Velocity – Time Graphs Remember: Slope indicates acceleration Area Under the Curve Indicates Displacement! Using slope formula, and equation for acceleration

can be derived.

if

if

tt

vv

t

va

SI Unit for acceleration is

m/s 2

Velocity – Time Graphs

Describe the motion of each object as represented on the velocity – time graph!

Vel

ocity

(m

/s)

Time (s)

3.2 Motion with Constant Acceleration Using the equations for average velocity and

average acceleration, we can come up with (derive) several equations for motion with uniform acceleration.

Depending on variables known or given, these equations can be used to solve for vf , vi , df , di , a, or t.

Velocity with Average Acceleration The final velocity is equal to the initial

velocity plus the product of the average acceleration and time interval.

This equation is simply the average acceleration equation rearranged.

atvv if

Position with Constant Acceleration An object’s position at a time after the initial time

is equal to the sum of its initial position, the product of the initial velocity and the time, and half the product of the acceleration and the square of the time.

This equation is derived from a velocity vs. time graph.

2

2

1attvdd iiif

An Alternative Expression Sometimes a time interval is not known and

we will still need to relate position, velocity, and acceleration.

Rearranging and substituting the last two equations will give us the equation

advv if 222

Summary of Equations

if

if

tt

vv

t

va

atvv if

advv if 222

2

2

1attvdd iiif

Making a “Given” and “Find” column when you do problems will help you decide which equation to use for each problem. PRACTICE is the best way to figure this out.

3.3 Free Fall

Galileo Galilei is credited with doing the first real studies on the effects of gravity.

His conclusion: neglecting the effect of the air, all objects in free fall has the same acceleration.

Substance, mass, or height of drop have no significant effect on acceleration due to gravity.

g = 9.8m/s2

Acceleration Due to Gravity Acceleration of an object in free fall that

results from the influence of Earth’s gravity. For each second in free fall, velocity will

increase by 9.8m/s. In each second, the distance will become

successively larger.

Positive or Negative

When analyzing free fall, treating acceleration as positive or negative depends on the coordinate system used.

If up = positive, then acceleration is –g If down = positive, then acceleration is +g

Ball Thrown Upward

If you thrown something upward and choose up as positive thenObject leaves hand with positive velocityAcceleration is downward so use –gVelocity and acceleration are in the opposite

direction, speed of ball decreases

What is the value for g at the top of the throw? Explain.

Graphing Free Fall

What would a Velocity-Time Graph look like for an object in free fall? (Up is positive.)

What would a Position-Time Graph of the same motion look like?