Unit 3, Uniform Acceleration Notes

1. Find the instantaneous velocity at t = 2.0 s and t = 8.0s by finding slopes of tangents. To find the slope of the tangent I will find the slope of the secants lines where the mid-times are 2 and 8s. 2. Determine the average acceleration from t = 2.0s to t = 8.0s

∆V = change in the instantaneous velocity (not the average velocity).

Notes v=

∆𝑿

∆𝒕 a=

∆𝒗

∆𝒕

0,25

4,21

6, 16

10, 0 8 2

Physics Bell Work, Wednesday, Sept 30

2. Draw a motion map for the runner shown above showing that velocity is not constant but acceleration is uniform .

v v

v v v v

a a a a a

a

v

a

Δt = 1s

Physics Bell Work, Wednesday, Sept

30

3. How are motion maps in questions 1 & 2 different? Map 1 is constant velocity (arrows same length, same dot spacing. Map 2 is increasing velocity (each arrow longer, dot spacing bigger).

Physics Bell Work, Monday, Feb 20

t (s) x (m)

0.0 4.0

2 10.0

4.0 16.0

6.0 22.0

8.0 28.0

10.0 34.0

4. Plot the data (draw the graph)

5. Calculate the slope 6. Determine the velocity. 5.

𝟑𝟒 𝒎 − 𝟒 𝒎

𝟏𝟎 𝒔= 𝟑

𝒎

𝒔

6. Slope of an x-t graph = velocity, 3 m/s v = 𝒙𝒇 − 𝒙𝒊

𝒕𝒇−𝒕𝒊

4

.

(s)

Worksheet

3, question

2

Bell Work Wednesday, 10/3/18 1. On a motion map how do we show an object is speeding up or slowing down? 2. Draw velocity and acceleration vectors that show an object that is speeding up and then slowing down to a stop at uniform acceleration.

Object speeding up

Object slowing down to a stop.

• When the object’s acceleration vectors are in the same direction as its velocity vectors, the object is speeding up.

• When the vectors are in opposite directions, the object is slowing down.

v a

v

a

We must add an acceleration vector to the motion map.

1. A ball rolling up a ramp and the reversing direction and rolling down the ramp. Draw the motion map, velocity-time and acceleration-time graph that describes this motion.

0

v

a

v (

m/s

)

+

-

a (

m/s

2)

Time (s)

-

+ a

v

If an object’s average acceleration during a time interval

is known, then it can be used to determine how much the

velocity changed during that time.

The definition of average acceleration:

Velocity with Average Acceleration

can be rewritten as follows:

Section 3.2-2

The equation for final velocity with average

acceleration can be written as follows:

Velocity with Average Acceleration

The final velocity is equal to the initial velocity

plus the product of the average acceleration and

time interval.

Section 3.2-3

In cases in which the acceleration is constant, the

average acceleration, ā, is the same as the

instantaneous acceleration, a. The equation for

final velocity can be rewritten to find the time at

which an object with constant acceleration has a

given velocity.

It also can be used to calculate the initial velocity

of an object when both the velocity and the time at

which it occurred are given.

Velocity with Average Acceleration

Section 3.2-4

On the graph shown on the

right, v is the height of the

plotted line above the t-axis,

while Δt is the width of the

shaded rectangle. The area of

the rectangle, then, is vΔt, or

Δd. Thus, the area under the

v-t graph is equal to the

object’s displacement.

Position with Constant Acceleration

Section 3.2-9

The area under the

v-t graph is equal to the

object’s displacement.

Position with Constant Acceleration

Section 3.2-10

Finding the Displacement from a

v-t Graph

The v-t graph shows the

motion of an airplane. Find

the displacement of the

airplane at Δt = 1.0 s and

at Δt = 2.0 s.

Section 3.2-11

Are the units correct?

Displacement is measured in meters.

Do the signs make sense?

The positive sign agrees with the graph.

Is the magnitude realistic?

Moving a distance of about one football field in 2 s

is reasonable for an airplane.

Finding the Displacement from a

v-t Graph

Section 3.2-22

Section

3.1 Acceleration

Average acceleration is equal to the change in

velocity, divided by the time it takes to make that

change.

The following equation expresses average

acceleration as the slope of the velocity-time

graph.

Determining Acceleration from a v-

t Graph

Section 3.1-43

Section

3.1 Section Check

Which of the following statements correctly defines

acceleration?

Question 1

A. Acceleration is the rate of change of displacement of

an object.

B. Acceleration is the rate of change of velocity of an

object.

C. Acceleration is the amount of distance covered in

unit time.

D. Acceleration is the rate of change of speed of an

object.

Section 3.1-44

Section

3.1 Section Check

Answer 1

Reason: The rate at which an object’s velocity

changes is called acceleration of the

object.

Section 3.1-45

Section

3.1 Section Check

What happens when the velocity vector and the

acceleration vector of an object in motion are in

the same direction?

Question 2

A. The acceleration of the object increases.

B. The speed of the object increases.

C. The object comes to rest.

D. The speed of the object decreases.

Section 3.1-46

Section

3.1 Section Check

Answer 2

Reason: When the velocity vector and the

acceleration vector of an object in

motion are in the same direction, the

speed of the object increases.

Section 3.1-47

Section

3.1 Section Check

On the basis of the

velocity-time graph of a

car moving up a hill, as

shown on the right,

determine the average

acceleration of the car?

Question 3

A. 0.5 m/s2

B. -0.5 m/s2

C. 2 m/s2

D. -2 m/s2

Section 3.1-48

Section

3.1 Section Check

Answer 3

Reason: Average acceleration of an object is

the slope of the velocity-time graph.

Section 3.1-49

Section

3.2 Motion with Constant Acceleration

If an object’s average acceleration during a time interval

is known, then it can be used to determine how much the

velocity changed during that time.

The definition of average acceleration:

Velocity with Average Acceleration

can be rewritten as follows:

Section 3.2-2

Section

3.2 Motion with Constant Acceleration

The equation for final velocity with average

acceleration can be written as follows:

Velocity with Average Acceleration

The final velocity is equal to the initial velocity

plus the product of the average acceleration and

time interval.

Section 3.2-3

Section

3.2 Motion with Constant Acceleration

In cases in which the acceleration is constant, the

average acceleration, ā, is the same as the

instantaneous acceleration, a. The equation for

final velocity can be rewritten to find the time at

which an object with constant acceleration has a

given velocity.

It also can be used to calculate the initial velocity

of an object when both the velocity and the time at

which it occurred are given.

Velocity with Average Acceleration

Section 3.2-4

Section

3.2 Motion with Constant Acceleration

The position data at

different time intervals

for a car with constant

acceleration are shown

in the table.

The data from the table

are graphed as shown

on the next slide.

Position with Constant Acceleration

Section 3.2-5

Section

3.2 Motion with Constant Acceleration

The graph shows that the

car’s motion is not uniform:

the displacements for equal

time intervals on the graph

get larger and larger.

The slope of a position-time

graph of a car moving with a

constant acceleration gets

steeper as time goes on.

Position with Constant Acceleration

Section 3.2-6

Section

3.2 Motion with Constant Acceleration

The slopes from the position

time graph can be used to

create a velocity-time graph

as shown on the right.

Note that the slopes shown

in the position-time graph

are the same as the

velocities graphed in the

velocity-time graph.

Position with Constant Acceleration

Section 3.2-7