Top Banner
Motion in One Direction Chapter 2
48

Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives: Describe motion in terms of frame of reference, displacement, time,

Jan 02, 2016

Download

Documents

Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Motion in One Direction

Chapter 2

Page 2: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

2-1: Displacement and Velocity

Main Objectives:Describe motion in terms of frame of

reference, displacement, time, and velocity.Calculate the displacement of an object

traveling at a known velocity for a specific time interval.

Construct and interpret graphs of position versus time.

Page 3: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Chapter 2 Introduction The motion of objects is an important part

of everyday lifeMotion was the first aspect of the physical

world to be thoroughly studied Our modern understanding of motion was

first established during the sixteenth and seventeenth centuriesGalileo Galilei (1564-1642) and Isaac Newton

(1642-1727) provided significant contributions to this understanding

Page 4: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Chapter 2 Introduction The study of the motion of objects, force,

and energy form the field called mechanicskinematics – the description of how objects

movedynamics – studies why objects move as they

do

Page 5: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Speed

average speed – the distance traveled divided by the time it takes to travel this distance

average speed =

Example: A car that travels 240km in 3h has an average speed of 80km/h.

distance traveled time elapsed

Page 6: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Converting Units

It is often necessary to convert between unitsFor example converting km/h to m/s

This can be accomplished by using the technique of dimensional analysis which was introduced in chapter one

Page 7: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Converting Units

When using dimensional analysis, one must make sure to:

1. Use correct conversion factors (must be equal to one), and

2. Set-up the problem to cancel out the appropriate units

Page 8: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Converting Units

Example: Convert 80km/h into m/s.

Practice: Convert 35.0m/s into km/h.

80km x 1m x 1h x 1min = 22m/s 1h 10-3km 60min 60s

Page 9: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Frame of Reference Every measurement must be made with

respect to a frame of reference – this provides the information necessary to understand the measurementThe frame of reference is a way of comparing

measurements Example: You are on a train that is traveling at a

speed of 80km/h. A passenger on the train walks by you at a speed of 5km/h.

Does the passenger have an average speed of 5km/h or 85km/h?

Page 10: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Frame of Reference

Both average speeds of the passenger are correct if given the proper frame of reference.The person has a speed of 5km/h relative to

the train.The person has a speed of 85km/h relative to

the ground. The frame of reference should be specified

whenever there might be confusion.

Page 11: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Frame of Reference

Other physical quantities are dependent on a frame of reference.Distance: Columbus, OH is 230km away from

ClevelandDirection: The plane was flying at 1000km/h

heading northeast from Cleveland.

Page 12: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Frame of Reference

In physics, a set of coordinate axes are used to represent a frame of reference.

Page 13: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Vectors and Scalars

vector – a quantity which has a direction as well as a magnitudeExamples: velocity, displacement, force,

momentum scalar – a quantity that only has a

magnitudeExample: time, temperature, energy

Page 14: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Average Velocity and Displacement

The terms speed and velocity are often used interchangeably, but in physics there is a distinctionvelocity – used to signify both the magnitude

(the numerical value) of how fast an object is moving and the direction the object is moving

Speed is just a magnitude

Page 15: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Average Velocity and Displacement

In addition, average velocity is defined in terms of displacement rather than total distance traveled

average velocity = displacement time elapsed

Page 16: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Average Velocity and Displacement

displacement (∆x or ∆y) – an object’s change in position

displacement = final position – initial position

∆x = x – x0 or ∆y = y – y0 Example: A person walks 70m to the east,

turns around, and walks back (west) 30m.The total distance traveled is 100m, but the

displacement is only 40m because the person is 40m away from the starting point

Page 17: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Average Velocity and Displacement

Example (cont.): Assume the walk in the following example took 80s. average speed = 100m/80s = 1.3m/s average velocity (magnitude) = 40m/80s = 0.50 m/s

This discrepancy between speed and the magnitude of velocity only occurs when dealing with average values

Page 18: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Average Velocity and Displacement

When an object moves from on point to another:Displacement = x – x0 = ∆xElapsed Time = t – t0 = ∆t

Therefore: v = =

The sign for displacement (+/-) can indicate the direction of motion

∆x x – x0

∆t t – t0

_

Page 19: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Average Velocity and Displacement

Example: The position of a runner as a function of time is plotted as moving along the x-axis of a coordinate system. During a 3.00s interval, the runner’s position changes from x0 = 50.0m to x = 30.5m. What was the runner’s average velocity?

Page 20: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Instantaneous Velocity

instantaneous velocity – the velocity at any instant of timeExample: If you travel along a straight road for

150 km in 2.0 h, the magnitude of your average velocity would be 75 km/h. However, it is unlikely that your velocity was 75km/h at every instant.

Page 21: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Instantaneous Velocity

An object with a uniform (constant) velocity over a particular time would have the same instantaneous velocity at any instant as its average velocity.

Page 22: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Position vs. Time Graphs

On a position-time graph, an object with a uniform velocity would for a straight line.The velocity (constant)

would be equal to the slope of the line.

Page 23: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Position vs. Time Graphs

An object without a uniform velocity would not form a straight line on a position-time graphThe instantaneous velocity of the object

would be equal to the slope of a tangent line at any point along the curve

Page 24: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Position vs. Time Graphs

Page 25: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Acceleration An object whose velocity is changing is

said to be accelerating.This can be a change in the magnitude of

velocity Example: A car’s velocity increasing from 0 to

80km/h

Or it can be a change in the object’s direction

A plane traveling at a constant 600km/h turns from north to northeast.

Page 26: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Acceleration average acceleration – the rate of change

of velocity It is the change in velocity divided by the time

taken to make this change

average acceleration =

a = =

change of velocity time elapsed

_ v – v0 ∆v t – t0 ∆t

Page 27: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Acceleration (Units) Usually the same unit of time is used when

expressing acceleration (the same unit of time for the velocity and for the ∆t).

= =

Complete the previous example using m/s for the velocity units.

m/s m m s s•s s2

Page 28: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Acceleration

Example: A car accelerates along a straight road from rest to 60km/h in 5.0s. What is the magnitude of its average acceleration?

Page 29: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

AccelerationExample: A car is moving along a straight highway

(which can be represented as right along an x-axis) and the driver puts on the brakes. If the initial velocity is v0 = 15.0 m/s and it takes 5.0 s to slow down to v = 5.0 m/s, what was the car’s average acceleration in m/s2?

Page 30: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Acceleration

Acceleration is a vector quantity – it has a magnitude and a direction

Page 31: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Uniform (Constant) Acceleration

Objects are said to have uniform acceleration when the magnitude of the acceleration is constant and the motion is in a straight line The relationships between position, velocity,

constant acceleration, and the time in motion can be expressed with various equations.

Page 32: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Uniform Acceleration

Variables:x0 = initial position x = position at time, t

v0 = initial velocity v = velocity at time, t

t = time elapsed

Page 33: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Uniform Acceleration - Equations

The average velocity of an object with uniform acceleration will be halfway between the initial and final velocities.

Equation 1: v = v0 + v 2

_

Page 34: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Uniform Acceleration - Equations

The final velocity of an object can be calculated when its acceleration, initial velocity, and the time elapsed is known.

Equation 2: v = v0 + at

Page 35: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Uniform Acceleration - Equations

The position of an object after a certain amount of time can be calculated when it is undergoing constant acceleration if you know the initial velocity.

Equation 3: x = x0 + v0t + ½ at2

Page 36: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Uniform Acceleration - Equations

The final velocity can be calculated without knowing the time elapsed if the initial velocity, displacement, and acceleration are known.

Equation 4: v2 = v02 + 2a(x - x0)

*Since a square root can be either + or -, you must determine the direction of motion by comparing the directions of the initial velocity and acceleration

Page 37: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Falling Bodies

A common example of uniformly accelerated motion is an object in free fall near the Earth’s surfacefree fall – the motion of a body when only the

force due to gravity is acting on the body Until Galileo’s time, it was believed that

heavier objects fall faster than lighter objects and that the speed of fall is proportional to how heavy the object is.

Page 38: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Falling Bodies

Why is it that a feather will take longer to hit the ground than an apple when dropped from the same height?

Page 39: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Falling Bodies

Galileo’s Hypothesis: at a given location on the Earth and in the absence of air resistance, all objects fall with the same uniform accelerationThis acceleration is called the acceleration

due to gravity on the Earth It is approximately: g = 9.80 m/s2, downward or

g = - 9.80 m/s2

Page 40: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Falling Bodies

In the absence of air resistance, all objects fall with the same constant acceleration regardless of their masses.

Page 41: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Falling Bodies

The effects of air resistance are often small and will be disregarded in most examples.However, the effects of air resistance are

noticeable even on reasonably heavy objects if the distance the object falls is very large

Page 42: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Falling Bodies

If an object falls far enough in air, it will reach a maximum velocity called terminal velocity.Terminal velocity is reached when the force of

air resistance (which increases with speed) reaches a magnitude equal to the force of gravity.

Page 43: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Falling Bodies

Free fall acceleration is constant during upward and downward motion.

Page 44: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Falling Bodies

Page 45: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Falling Bodies

You can use the equations for objects undergoing uniformly accelerated motion to solve problems involving free falling objects In these situations, “a” will be equal to

g = - 9.80 m/s2 near the Earth’s surface Also, since the motion is vertical, x will be

replaced with y and x0 is replaced with y0.

Page 46: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Example 1:

Suppose that a ball is dropped from a tower 70.0 m high. How far will it have fallen after 1.00, 2.00, and 3.00 s? Assume y is negative with downward motion.

Page 47: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Example 2:

Suppose the ball in the previous example was thrown downward with a speed of

-3.00 m/s instead of being dropped. (a) What would be the ball’s position after 1.00 s and after 2.00 s? (b) What would its speed after 1.00 s and 2.00 s. Compare these speeds to the speeds of a ball that was dropped (v0 = 0).

Page 48: Motion in One Direction Chapter 2. 2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time,

Displacement & Velocity vs. Time Graphs