Chapter 2, Kinematics

Terminology • Mechanics = Study of objects in motion.

– 2 parts to mechanics.

• Kinematics = Description of HOW objects move. – Chapters 2 & 4 (Ch. 3 is mostly math!)

• Dynamics = WHY objects move.– Introduction of the concept of FORCE.

– Causes of motion, Newton’s Laws

– Most of the course after Chapter 4

For a while, assume ideal point masses (no physical size).

Later, extended objects with size.

Brief Overview of the Course • Translational Motion = Straight line motion.

– Chapters 2,4,5,9,13

• Rotational Motion = Moving (rotating) in a circle.– Chapters 6,10,11,12

• Oscillations = Moving (vibrating) back & forth in same path.– Chapter 15

Continuous Media• Waves, Sound

– Chapters 16,17,18 • Fluids = Liquids & Gases

– Chapter 14

Conservation Laws: Energy, Momentum, Angular Momentum

– Just Newton’s Laws expressed in other forms!

THE THEME OF THE COURSE IS NEWTON’S LAWS

OF MOTION!!

Motion in One Dimension

1. Displacement:

Sects. 1-4: Outline

2. Average velocity:

f ix x x

xv

t

3. Instantaneous

velocity:0t

x dxv Lim

t dt

4. Speed is the absolute value of the velocity

2. Time interval: f it t t

5. Average acceleration:

f iv va

t

6. Instantaneous

acceleration: 0t

v dva Lim

t dt

Reference Frames, Coordinate Systems &

Displacement • Every measurement must be made with respect

to a reference frame.

• Usually, speed is relative to the Earth.

Coordinate Axes • Define reference frame using a standard

coordinate axes.

• 2 Dimensions (x,y)

• Note, if its convenient,

could reverse + & - !

Coordinate Axes • 3 Dimensions (x,y,z)

• Define direction using these.

Displacement & Distance • Distance traveled by an object

displacement of the object!

• Displacement = change in position of object.• Displacement is a vector (magnitude & direction). Distance is a

scalar (magnitude).• Figure: distance = 100 m, displacement = 40 m East

Displacement

x1 = 10 m, x2 = 30 m

Displacement ∆x = x2 - x1 = 20 m

∆ Greek letter “delta” meaning “change in”

x1 = 30 m, x2 = 10 m

Displacement ∆x = x2 - x1 = - 20 m

Displacement is a VECTOR

Vectors and Scalars • Many quantities in physics, like displacement,

have a magnitude and a direction. Such quantities are called VECTORS.– Other quantities which are vectors: velocity,

acceleration, force, momentum, ...

• Many quantities in physics, like distance, have a magnitude only. Such quantities are called SCALARS.– Other quantities which are scalars: speed,

temperature, mass, volume, ...

• The Text uses BOLD letters to denote vectors.

• I usually denote vectors with arrows over the symbol.

• In one dimension, we can drop the arrow and remember that a + sign means the vector points to right & a minus sign means the vector points to left.

Average Velocity Average Speed (Distance traveled)/(Time taken)

Average Velocity (Displacement)/(Time taken)

• Velocity: Both magnitude & direction describing how fast an object is moving. A VECTOR. (Similar to displacement).

• Speed: Magnitude only describing how fast an object is moving. A SCALAR. (Similar to distance).

• Units: distance/time = m/s

Scalar→

Vector→

Average Velocity, Average Speed • Displacement from before. Walk for 70 s.

• Average Speed = (100 m)/(70 s) = 1.4 m/s

• Average velocity = (40 m)/(70 s) = 0.57 m/s

• In general:

∆x = x2 - x1 = displacement∆t = t2 - t1 = elapsed timeAverage Velocity:

(∆x)/(∆t) = (x2 - x1)/(t2 - t1)

Example • Person runs from x1 = 50.0 m to x2 = 30.5 m

in ∆t = 3.0 s. ∆x = -19.5 m

Average velocity = (∆x)/(∆t) = -(19.5 m)/(3.0 s) = -6.5 m/s. Negative sign indicates DIRECTION, (negative x direction)

Instantaneous velocity

velocity at any instant of time

average velocity over an

infinitesimally short time

• Mathematically, instantaneous velocity:

v = lim ∆t 0 [(∆x)/(∆t)] ≡ (dx/dt)

lim ∆t 0 ratio (∆x)/(∆t) considered as a whole for smaller & smaller ∆t. As you should know, mathematicians call this a derivative.

Instantaneous velocity

v ≡ time derivative of displacement x

Acceleration • Velocity can change with time. An object whose

velocity is changing with time is said to be accelerating.

• Definition: Average acceleration = ratio of change in velocity to elapsed time.

a (∆v)/(∆t) = (v2 - v1)/(t2 - t1)

– Acceleration is a vector.

• Instantaneous acceleration

a = lim ∆t 0 [(∆v)/(∆t)]

• Units: velocity/time = distance/(time)2 = m/s2

Example

Conceptual Question

1. Velocity & acceleration are both vectors.

Are the velocity and the acceleration

always in the same direction?

NO!!

If the object is slowing down, the acceleration vector is in the opposite direction of the velocity vector!

Example

Deceleration

• “Deceleration”: A word meaning “slowing down”. We try to avoid using it in physics. Instead (in one dimen.) talk about positive & negative acceleration.

• This is because (for one dimen. motion) deceleration does not necessarily mean the acceleration is negative!

Conceptual Question

2. Velocity & acceleration are vectors.

Is it possible for an object to have a zero

acceleration and a non-zero velocity?

YES!!

If the object is moving at a constant velocity, the acceleration vector is zero!

Conceptual Question3. Velocity & acceleration are vectors.

Is it possible for an object to have a zero

velocity and a non-zero acceleration?

YES!!

If the object is instantaneously at rest (v = 0) but

is either on the verge of starting to move or is

turning around & changing direction, the

velocity is zero, but the acceleration is not!

Examples

Example 2.1: Calculating Average Velocity & Speed, p. 28

Problem: Use the figure & table to find the displacement & the average velocity of the car between positions (A) and (F).

Example 2.3: Average & Instantaneous Velocity, p. 25 Problem: A particle moves along the x axis. Its x coordinate varies with time as x = -4t + 2t2. x is in meters & t is in seconds. The position-time graph for this motion is in the figure.

a) Determine the displacement of the particle in the time intervals t = 0 to t = 1 s & t = 1 s to t = 3 s (A to B & B to C)

b) Calculate the average velocity in the time intervals t = 0 to t = 1 s & t = 1 s to t = 3 s (A to B & B to C)

c) Calculate the instantaneous velocity of the particle at t = 2.5 s (point C) .

Example 2.4: Graphical Relationships between x, vx (v) & ax (a) Problem: The position of an object moving along the x axis varies with time as in the figure. Graph the velocity versus time and acceleration versus time curves for the object.

Example 2.5, Average & Instantaneous Acceleration, p. 30

Problem: The velocity of a particle moving along the x axis varies in time according to the expression v = (44 - 10t 2), where t is in seconds.

a) Find the average acceleration in the time interval t = 0 to t = 2.0 s.

b) Find the acceleration at t = 2.0 s.

Ch. 2, Sects. 1-4: Motion in One Dimension

1. Displacement:

Summary

2. Average velocity:

f ix x x

xv

t

3. Instantaneous

velocity:0t

x dxv Lim

t dt

4. Speed is the absolute value of the velocity

2. Time interval: f it t t

5. Average acceleration:

f iv va

t

6. Instantaneous

acceleration: 0t

v dva Lim

t dt