Physics

CHAPTER TWO

Motion in One Dimension

Teacher: Luiz Izola

Chapter Preview

1. Position, Distance, and Displacement

2. Average Speed and Velocity

3. Instantaneous Velocity

4. Acceleration

5. Motion with Constant Acceleration

6. Applications of Motion Equations

7. Freely Falling Objects

Learning Objectives

How to analyze one-dimensional motion related to displacement, time, speed, and velocity.

How to distinguish between accelerated and non-accelerated motion.

Introduction

Mechanics is the study of how objects move, how they respond to external forces, and how other factors, such as size, mass, mass distribution affect their motion. Kinematics, from Greek kinema, means motion. It is the study of motion and how to describe it, without regard for how the motion was caused.

Displacement

Simplest form of motion: One-dimensional

Frame of reference: In order to analyze any motion related problem, we need to have a point of reference in order to create logical assumptions.

Displacement: Length from initial position to final on a straight line. SI unit is meter

Δx = xf – xi

Displacement is not always distance because displacement can be +/- .

Practice Session

1. Heather and Mathew walk with an average velocity of 0.98m/s eastward. If it takes them 34 min to walk to the store, what is their displacement.

2. The distance from your house to the library is 5 miles. From the library to the park is 2 miles. You walked to the library, then to the park and all the way back to your house. Which is the displacement and distance values?

3. If it takes you 9.5 minutes to walk with an average velocity of 48.0 km/h to the east. How long it will take you to drive 144 km in a straight highway?

Displacement and Velocity

Displacement can be positive or negative, it depends on the direction of the motion.

Velocity – Displacement divided by the time interval during the displacement occurrence.

Average Velocity = (xf - xi)/(tf – ti) = Δx / Δt

Where: Δx = change in position

Δt = change in time

Displacement and Velocity

Velocity SI unit is: meters/seconds.

Velocity can be positive or negative, depending on sign of the displacement.

Time is always positive.

Average velocity is equals to the constant velocity needed to cover the given displacement in a time interval.

Practice Session

1. During a race on level ground, Andra runs with an average velocity of 6.02m/s to the east. What is Andra’s displacement after 137 seconds.

2. You drive 4 mi at 30 mi/h and then another 4 mi at 50 mi/h. Is your average speed for the 8 miles:

(a) > 40 mi/h (b) = 40 mi/h (c) < 40 mi/h

Velocity and Speed

Velocity is not the same as speed: Speed does not take into consideration direction.

Velocity can be interpreted by graphs. See Figs. 2.6, page 38.

Slope of line is related to average velocity:

Slope = Δy / Δx, where:

Δy = Vertical coordinate change

Δx = Horizontal coordinate change

Acceleration

Acceleration is the rate of velocity change with respect to time. Average acceleration is calculated by dividing the total velocity change by the time change.

aavg = Δv / Δt

SI acceleration units: meters / second2

Practice Session

1. A child is pushing a shopping cart at a speed of 1.5 m/s. How long it will take for him/her to push the cart through 9.3 meters?

2. A swimmer swims from the north to the end south of a swimming pool (50m) in 20.0 seconds and makes the return trip in 20.0 seconds. What is the swimmer average velocity?

3. Two students walk in the same direction along a straight path, at a constant speed – one at 0.90m/s and the other at 1.90m/s. They start at the same point and time.(a) How much sooner the faster student arrives at a destination 780 meters away?(b) How far they have to walk so the faster arrives 5.5 minutes ahead of the slower?

Acceleration

Since acceleration is directly proportional to velocity, it has direction and magnitude.

Velocity positive Acceleration positive

Velocity negative Acceleration negative

Velocity constant Acceleration is zero

Acceleration

A negative acceleration does not mean that speed is decreasing. It is related to direction.

When velocity and acceleration of an object have the same sign, object’s speed increases. Velocity and acceleration point to same direction.

When they have opposite signs, speed decreases. Velocity and acceleration point to opposite directions.

Practice Session

1. A shuttle bus slows down with an average acceleration of -1.8m/s2. How long does it take the bus to slow down from 9.0m/s to a complete stop?

2. Let us try example 2.3, page 41.

3. Let us try example 2.4, page 42.

4. Let us try example 2.5, page 44.

Displacement with Constant Acceleration

FORMULAS:

v = vo + aΔt

x = xo + ½ (v + vo) Δt

x = xo + viΔt + ½ a(Δt)2

vf2 = vi

2 + 2a Δx

Practice Session

1. A plane starting at rest at one end of a runway undergoes a uniform acceleration of 4.8m/s2 for 15 seconds before takeoff. What is the speed at takeoff? How long must the runway be for the plane to be able to take off?

2. Let us try example 2.6, page 46.

3. Let us try example 2.7, page 47.

Practice Session

1. A person pushing a stroller from rest, uniformly accelerating at a rate of 0.500 m/s2. What is the velocity of the stroller after it has traveled 4.75 m?

2. Let us try example 2.8, page 48.

Falling Objects

Free-falling bodies have constant acceleration.

If air resistance is disregarded, objects dropped near the surface of the planet fall with the same constant acceleration. This acceleration is due to gravity, and the motion is referred to as Free Fall.

Falling Objects

Gravity’s Acceleration: ag = 9.80 m/s2

Because an object in free fall is acted on only by

gravity, ag is also known as free-fall acceleration.

Acceleration is constant during upward and downward motions.

Let try 2.9, page 52, 2.10 and 2.11, page 53.

Practice Session

Jason hits a volleyball so that it moves with an initial velocity of 6.0 m/s straight upward. If the volleyball starts from 2.0 m above the floor. How long it will be in the air before it strikes the floor?

Let us try page 56: 2.12.

Homework

Page 57: 11, 15, 17, 19

Page 58: 21, 23, 25, 27, 29

Page 59: 31, 33, 35, 39, 41, 43

Page 60: 47, 49, 51, 53, 55, 57, 59, 61

Page 61: 71, 73, 75, 77,79, 83, 85, 87

Motion Formulas vf

2 = vi2 + 2a Δx

Δx = viΔt + ½ a(Δt)2

vf = vi + aΔt

Δx = ½ (vi + vf) Δt

aavg = Δv / Δt