Velocity Time Graphs

What do Position Time graphs show?

• They show how position changes with time.

• So far we have studied graphs that show uniform motion. Or a constant velocity.

• We have seen some graphs that change slope, but they do this instantaneously.

• This is an unrealistic way to view the world.

What does a changing d t graph look like?

• This we don’t have the tools to work with a graph like this it varies too much.

What kinds of changing d t graphs can we handle?

• Consistent curves.

How do we read a curve on a d t graph?

• Look at how the slope is changing.• The best way to do this is to find the slope

at several points along the curve.• The slope of a curve at any point can be

found by drawing a tangent at that point.

The slope of a Tangent

• Tangent is any line that just touches the curve at that point.

• Looking at the tangent line you can tell that the slope is increasing from point 1 to point 3

1

2

3

What does a changing slope on a d t graph mean?

• Slope on a d t graph is the velocity of the object.

• So a changing slope means a changing velocity!

This graph has an increasing slope so that means the velocity is increasing or has a positive change.

This graph has a decreasing slope so that means the velocity is decreasing or has a negative change.

What about these?

• These can blow your mind unless you apply the tangent model.

This graph starts off with a steep negative slope and goes to a shallower slope. The velocity has a high negative value and goes to a smaller negative value. This is a positive change in velocity.

This graph starts off with a shallow negative slope and goes to a steep negative slope. The velocity has a low negative value and goes to a high negative value. This is a negative change in velocity.

Now what?

• These curved graphs are fine to understand the motion qualitatively however, they are not very useful quantitatively.

• Since these graphs are continuously or consistently changing. Then they too, are a form of uniform motion. A uniformly changing motion.

• Lets see how a d t graph translates into a v t graph.

Exploring a v t Graph• What does a linear d t graph

look like as a v t?

v→(m/s)

t (s)

4

5

d→(m)

t (s)

20

5

Slope = 4 m/s

A constant slope means a constant or unchanging velocity. That translates into a horizontal v t graph.

A changing d t graph

• If the d t graph is uniformly changing in the positive direction.• You can tell the change by looking at how the tangent changes!• Then the velocity is consistently changing.• Which translates into a positively sloped straight line.

Positively changingvelocity

Positive acceleration

D (m)

V (m/s)

Horizontal = 0 velocity

Steep slope = high velocity

Starting at 0 velocity

Ending at a high velocity

A changing d t graph

• If the d t graph is uniformly changing in the positive direction.• Then the velocity is consistently changing.• Which translates into a positively sloped straight line.• When velocity reaches maximum value it remains constant• Constant velocity means 0.00 acceleration = horizontal line

Positively changingvelocity

Positive acceleration

D (m)

V (m/s)

Constant velocity

Constant velocity

A changing d t graph

• If the d t graph is uniformly changing in the negative direction.• You can tell the change by looking at how the Tangent changes!• Then the velocity is consistently changing negatively.• Which translates into a negatively sloped straight line.

Negatively changingvelocity

Negative accelerationD

(m)

V (m/s)

Vertical = High velocity

Low slope = low velocity

Starting at a high velocity

Ending at a low velocity

A negatively changing d t graph

• If the d t graph is uniformly changing in the negative direction.

• You can tell how it is changing by looking at the Tangent.• Then the velocity is consistently changing.• Which translates into a negatively sloped straight line.

XX

Yes, we have a straight line again!

• We have a burning desire to calculate the slope.

• Slope of a v t graph would be….• Slope = rise = ∆v

run ∆t• This value will measure the rate of change

of velocity with time.• This is known as acceleration!

Exploring Acceleration

• This term describes how velocity changes with time.

• Acceleration is a vector quantity.• Units: m m x 1 m

s = s s = s2

s• What does 5.0 m/s2 mean?• It means that the velocity changes by + 5.0 m/s

every second.

What about deceleration?

• Avoid using this term altogether!• Why?• Deceleration indicates slowing down

toward zero velocity. • This would also inherently imply a

negative velocity.• This is not always true!

Lets See

• This is deceleration on a v t graph.• This object has a decreasing velocity

ending at zero.

What happens here?

• If we extend the deceleration graph beyond the zero velocity the object has an increasing velocity in the negative direction.

• This is NOT what deceleration means.

Deceleration

Acceleration in the negative direction.

What happens here?

• If there is a negative velocity to decelerate the object needs to have a positive change in velocity to reach zero velocity. This is a positive deceleration which doesn’t make sense!

Deceleration (positive)

Acceleration

Solution

• Always use acceleration: the rate of change of velocity with time. A vector quantity.

Acceleration in the negative direction. - a

Acceleration in the positive direction. + a

Assignment

• Complete V T handout assignment