Linear Motion: Velocity and Acceleration

Relative Motion • Everything moves, even things at rest • Relative – regarded in relation to something

else; depends on point of view, or frame of reference

• A book at “rest”, relative to you, is actually moving at 30 km/s with respect to the sun, and even faster the center of the galaxy

• Unless stated otherwise, when we discuss speeds of things in our environment, we mean speed with respect to the surface of Earth

• Motion is relative!

Motion is relative!

Speed • Speed – The measure of how fast something is

moving; the rate at which distance is covered. •  Instantaneous Speed – The speed at any

instant in time • Average Speed – The total distance covered

divided by the total time

Measured in meters/second [m/s]

Average vs. Instantaneous Speed

Which of the following are speeds? (Hint: There may be more than one correct answer!)

a. 5 m b. 10 m/s c. 4567 mph d. 0.009 s e. 17 miles f. 0.023 km/hr g. 12 cm/min

Check for Understanding

a. 5 m b. 10 m/s c. 4567 mph d. 0.009 s e. 17 miles f. 0.023 km/hr g. 12 cm/min

Velocity

• Velocity – Speed in a given direction ▫  Velocity can be (+) or (-)

• Constant Velocity – Requires that both constant speed and constant direction must be maintained

• Changing Velocity – Either the speed or the direction changes, so the velocity changes

Measured in meters/second [m/s]

• Average Velocity – change in position/change in time.

•  **Note: Average speed is different from average velocity. ▫  Average speed = distance traveled/time ▫  Average velocity = change in position/change in

time

€

vavg =ΔxΔt

vavg= average speed [m/s] Δ = change x = position [m] t = time [s]

•  The speedometer in every car also has an odometer that records the distance traveled. ▫  If the odometer reads zero at the beginning of a

trip and 35 km a half hour later (assuming the car is traveling forward in a straight line), what is the average velocity?

Check for Understanding

What is the difference between speed and velocity?

Speed is always (+). Speed does not have direction.

Velocity can be (+) or (-). Velocity has direction.

Speed = 5 m/s Velocity = -5 m/s

Acceleration • Acceleration – The rate at which velocity is

changing •  Term applies to both increases and decreases in

velocity (difference between positive and negative acceleration)

Measured in meters/second² (m/s²)

aavg= average acceleration[m/s2] Δ = change v = velocity[m/s] t = time [s]

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aavg =ΔvΔt

• Acceleration applies to changes in direction also; when the direction changes, the acceleration changes

• Most of the time we will concern ourselves with motion in a straight line, and can look at the change in speed.

• Gravity – The acceleration that causes objects to move towards the Earth or other large objects

Gravity = g = 9.8 m/s² ≈ 10 m/s2

•  In 1977 off the coast of Australia, the fastest speed by a vessel on the water was achieved. If this vessel were to undergo an average acceleration of 1.80 m/s2, it would go from rest to its top speed in 85.6 s. What was the speed of this vessel?

Velocity vs. Acceleration

Observe the animation of the three cars below. Which car or cars (red, green, and/or blue) are undergoing an acceleration? Study each car individually in order to determine the answer.

If necessary, review the definition of acceleration.

In Summary… •  Speed and __________ are different because

_____________ is a measure of how fast something is moving, and velocity is a measure of how fast something is moving in a certain ___________.

• Average __________ is the rate at which the velocity is changing with time.

• ___________ is the change in position divided by the change in time.

•  The units of acceleration are __________.

Check for Understanding

Linear Motion: Part II Chapter 2

FLT: I can solve word problems using the equations for linear motion with constant acceleration.

What we already know: • Velocity tells us how fast and in what direction

an object is moving. • Acceleration tells us the rate at which the

velocity is changing.

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vavg =ΔxΔt

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aavg =ΔvΔt

a= acceleration [m/s2] v0 = initial velocity [m/s] v = final velocity t = time [s]

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v = at + v0

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aavg =ΔvΔt

Example 1 • A runner is initially moving at 0.5 m/s and

accelerates at the rate of 0.9 m/s2 for 6 s. What is the velocity of the runner after the 6 s?

How far does an object move? •  The position of an object is related to the

acceleration, time, initial velocity, and initial position of the object.

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x =12at 2 + v0t + x0

x or y = position [m] x0 or y0 = initial position [m] a = acceleration [m/s2] v0 = initial velocity [m/s] t = time [s]

Example 2 • A ball is rolling across a table with an initial

velocity of 5 m/s. It accelerates at 2 m/s2. If the ball rolls for 3 seconds, how much distance does it cover?

Example 3 • A ball moving with a speed of 2.00 m/s increases

speed uniformly, so that in 40 s it has traveled 70.2 m. What is the magnitude of the ball’s acceleration?

One Last Equation… •  If you are given a problem where you are not

give time, and you are not asked to find time, use:

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v 2 − v02 = 2a(x − x0)

x = position [m] x0 = initial position [m] a = acceleration [m/s2] v0 = initial velocity [m/s] v = velocity [m/s]

Example • A radio-controlled toy car increases speed over a

distance 0f 15.2 m. If the car starts at rest and has a final speed of 0.76 m/s, what is the magnitude of its acceleration?

Linear Motion Equations

1.

2.

3.

4.

€

v 2 − v02 = 2a(x − x0)

€

x =12at 2 + v0t + x0€

v = at + v0

€

vavg =ΔxΔt

Linear Motion: Free Fall Chapter 2

FLT: I can solve free fall problems using the 3 equations for linear motion with constant acceleration.

• Free Fall – Objects that are only affected by gravity (neglecting air resistance)

• Elapsed Time – The time that has passed since the beginning of the fall

• We can find out about the motion of objects in free fall using the four linear motion equations.

Free Fall

Linear Motion Equations

1.

2.

3.

4.

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v 2 − v02 = 2a(x − x0)

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x =12at 2 + v0t + x0€

v = at + v0

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vavg =ΔxΔt

Acceleration due to gravity, g

•  If an object is in free fall (aka falling through the air towards the earth), it ALWAYS accelerates at ‘g’. (Remember, g = 10 m/s2)

• Gravity pulls objects towards earth (down), therefore we use acceleration due to gravity a = -g when we are solving free fall problems

•  In other words, a = -10 m/s2 for ANY object in free fall – ALWAYS!

• Miss Stein drops a ball off the roof of the gym. What is its acceleration?

• Miss Stein throws a tomato into the air. What is its acceleration?

Check for Understanding

-10 m/s2

-10 m/s2

Keys for solving free fall problems: 1.  Draw a picture! 2.  a = -10 m/s2

For objects moving up then down (as below) 1.  v at top = 0 m/s 2.  t to top = 1/2 (total t)

Example 2 • A pumpkin is released from rest at the top of the

gym, which is 271 m tall. Disregarding air resistance, calculate the displacement of the pumpkin after 2 s.

Example • Miss Stein throws an orange straight up into the

air with a speed of 60 m/s. The orange is in the air for 12 seconds. How high does the orange rise?

Air Resistance

•  Air resistance is responsible for the differences in accelerations that we see between an elephant and a feather. •  With a lack of air, these two items would fall at

the same rate!! •  Air resistance less noticeably effects more

condense objects (i.e. baseballs and stones)

The Elephant and the Feather, Air Drag