Kinematics

The branch of mechanics that studies the motion of a body without caring about what caused the motion.

Some Physics Quantities

Vector - quantity with both magnitude (size) & direction

Scalar - quantity with magnitude only

Vectors:

• Displacement
• Velocity
• Acceleration
• Momentum
• Force

Scalars:

• Distance
• Speed
• Time
• Mass
• Energy

Mass vs. Weight

On the moon, your mass would be the same, but the magnitude of your weight would be less.

Mass

• Scalar (no direction)
• Measures the amount of matter in an object

Weight

• Vector (points toward center of Earth)
• Force of gravity on an object

Vectors

• The length of the arrow represents the magnitude (how far, how fast, how strong, etc, depending on the type of vector).

• The arrow points in the directions of the force, motion, displacement, etc. It is often specified by an angle.

Vectors are represented with arrows

42°

5 m/s

Units

Quantity . . . Unit (symbol)

• Displacement & Distance . . . meter (m)
• Time . . . second (s)
• Velocity & Speed . . . (m/s)
• Acceleration . . . (m/s²)
• Mass . . . kilogram (kg)
• Momentum . . . (kg · m/s)
• Force . . .Newton (N)
• Energy . . . Joule (J)

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Units are not the same as quantities!

Kinematics definitions

• Kinematics – branch of physics; study of motion
• Position (x) – where you are located
• Distance (d ) – how far you have traveled, regardless of direction
• Displacement (Δx) – where you are in relation to where you started

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Slide 1-3

REPRESENTING MOTION

Four Types of Motion We’ll Study

Making a Motion Diagram

Examples of Motion Diagrams

The Particle Model

A simplifying model in which we treat the object as if all its mass were concentrated at a single point. This model helps us concentrate on the overall motion of the object.

Position and Time

The position of an object is located along a coordinate system.

At each time t, the object is at some particular position. We are free to choose the origin of time (i.e., when t = 0).

Slide 1-17

Particle

• Has position and mass.
• Has NO size or volume.
• Located at one point in space.

Position

• Location of a particle in space.
• One dimension (x)
• Two dimensions (x,y)
• Three dimensions (x,y,z)

0

1

2

3

-1

X (m)

1-Dimensional Coordinates

x = 1 m

Distance

• The total length of the path traveled by an object.
• Does not depend upon direction.
• “How far have you walked?”

0

1

2

3

-1

X (m)

1-Dimensional Coordinates

x_i = 1 m

Distance moved by particle is 2 meters.

Displacement

• The change in position of an object.
• Depends only on the initial and final positions, not on path.
• Includes direction.
• “How far are you from home?”

Displacement

• Represented by Δx.
• Δx = x₂ - x₁

where

x₂ = final position

x₁= initial position

0

1

2

3

-1

X (m)

1-Dimensional Coordinates

x_i = 1 m

Distance moved by particle is 2 meters.

Displacement of particle is -2 meters.

Distance vs Displacement

Checking Understanding

Maria is at position x = 23 m. She then undergoes a displacement ∆x = –50 m. What is her final position?

• –27 m
• –50 m
• 23 m
• 73 m

Answer

Maria is at position x = 23 m. She then undergoes a displacement ∆x = –50 m. What is her final position?

• –27 m
• –50 m
• 23 m
• 73 m

Checking Understanding

Two runners jog along a track. The positions are shown at 1 s time intervals. Which runner is moving faster?

Two runners jog along a track. The positions are shown at 1 s time intervals. Which runner is moving faster?

Answer

A

Checking Understanding

Two runners jog along a track. The times at each position are shown. Which runner is moving faster?

3. They are both moving at the same speed.

Two runners jog along a track. The times at each position are shown. Which runner is moving faster?

3. They are both moving at the same speed.

Answer

Average Speed

s_ave = d

t

Where:

s_ave = rate (speed)

d = distance

t = elapsed time

Average Velocity

v_ave = ∆x

∆t

Where:

v_ave = average velocity

∆x = displacement (x₂-x₁)

∆t = change in time(t₂-t₁)

Velocity vs Speed

• Average speed is always positive.
• Average velocity can be positive or negative depending direction.
• Absolute value of velocity can be used for speed if the object is not changing direction.

Average Velocity

V_ave = Δx/Δt, or the slope of the line connecting A and B.

Average Velocity

V_ave = Δx/Δt; still determined by the slope of the line connecting A and B.

Instantaneous Velocity

Determined by the slope of the tangent to a curve at a single point.

B

Acceleration

• A change in velocity is called acceleration.
• Acceleration can be
• speeding up
• slowing down
• turning

Uniformly Accelerated Motion

• In Physics B, we will generally assume that acceleration is constant.
• With this assumption we are free to use this equation:

a = ∆v

∆t

Units of Acceleration

The SI unit for acceleration is m/s².

Sign of Acceleration

Acceleration can be positive or negative.

The sign indicates direction.

General Rule

If the sign of the velocity and the sign of the acceleration is the same, the object speeds up.

If the sign of the velocity and the sign of the acceleration are different, the object slows down.

Velocity & Acceleration Sign Chart

Accelerating objects…

t

x

Note: each of these curves has many different slopes (many different velocities)!

Pick the constant velocity graph(s)…

A

t

x

C

t

v

B

t

x

D

t

v

Another accelerating object.

t

x

The tangent touches the curve at one point. Its slope gives the instantaneous velocity at that point.

Another tangent. Another instantaneous velocity!

Summary:�Constant position graphs

Summary:�Constant velocity graphs

Summary:�Constant acceleration graphs

Here is a motion diagram of a car moving along a straight stretch of road:

Which of the following velocity-versus-time graphs matches this motion diagram?

Checking Understanding

Here is a motion diagram of a car moving along a straight stretch of road:

Which of the following velocity-versus-time graphs matches this motion diagram?

Answer

Checking Understanding

A graph of position versus time for a basketball player moving down the court appears like so:

Which of the following velocity graphs matches the above position graph?

A graph of position versus time for a basketball player moving down the court appears like so:

Which of the following velocity graphs matches the above position graph?

Answer

A graph of velocity versus time for a hockey puck shot into a goal appears like so:

Which of the following position graphs matches the above velocity graph?

Checking Understanding

A graph of velocity versus time for a hockey puck shot into a goal appears like so:

Which of the following position graphs matches the above velocity graph?

Answer

Summary

v = v_o + at

x = x_o + v_ot + 1/2 at²

v² = v_o² + 2a(∆x)

Free Fall

• Occurs when an object falls unimpeded.
• Gravity accelerates the object toward the earth.

Acceleration due to gravity

• g = 9.8 m/s² downward.
• a = -g if up is positive.
• acceleration is down when ball is thrown up EVERYWHERE in the balls flight.

Summary

v = v_o - gt

x = x_o + v_ot - 1/2 gt²

v² = v_o² – 2g(∆x)

Symmetry

• When something is thrown upward and returns to the thrower, this is very symmetric.
• The object spends half its time traveling up; half traveling down.
• Velocity when it returns to the ground is the opposite of the velocity it was thrown upward with.
• Acceleration is –9.8 m/s² everywhere!