Linear Motion

Chapter 2

Vectors vs Scalars

• Scalars are quantities that have a magnitude, or numeric value which represents a size i.e. 14m or 76mph.

• Vectors are quantities which have a magnitude and a direction, for instance 12m to the right or 32mph east.

Describing how far you’ve gone

• Distancelength of the path between two pointsΔx

• Scalar• Standard units are

meters• A measure of how far

you have moved with respect to you (what a pedometer would measure)

• Displacementlength of the shortest path between two pointsΔx

• Vector• Standard units are

meters accompanied by direction.

• A measure of how far you are with respect to where you started (or change in position).

Distance vs Displacement

• The person, according to a pedometer has walked a total of 12m. That is the distance traveled.

• The person walking starts where she stops, so her displacement is zero.

Distance vs. Displacement

Start

End

6m

3m

3m

1m

Distance-Add all the distances together, totals 13m.

Add the left/right pieces and the up/down pieces and use the Pythagorean Theorem.

Displacement-Measured from beginning to end.

Distance vs. Displacement

Start

End

6m

3m

3m

1m

6m right + 3m left=3m right

3m down + 1m down=4m down

The total displacement is 5m.

You also need to include a direction, but we will take care of that in the next chapter.

Measuring how fast you are going

• Speedv• Scalar• Standard unit is m/s

• Velocityv• Vector• Standard unit is m/s,

plus direction

t

x

time

ntdisplacemev

t

x

time

distancev

Velocity and Speed

• If it take the person 4 seconds to walk around the square, what is her average speed and average velocity?

• For speed, Δx=12m and t=4s, so v=3m/s

• For velocity, Δx=0 and t=4s, so v=0m/s

Practice Problem

• A boy takes a road trip from Philadelphia to Pittsburgh. The distance between the two cities is 300km. He travels the first 100km at a speed of 35m/s and the last 200km at 40m/s. What is his average speed?

Practice Problem

• A boy takes a road trip from Philadelphia to Pittsburgh. The distance between the two cities is 300km. He travels the first 100km at a speed of 35m/s and the last 200km at 40m/s. What is his average speed?

Different types of velocity and speed

• Average velocity/speed• A value summarizing

the average of the entire trip.

• All that’s needed is total displacement/distance and total time.

• Instantaneous velocity• A value that

summarizes the velocity or speed of something at a given instant in time.

• What the speedometer in you car reads.

• Can change from moment to moment.

Acceleration

t

vv

t

vonacceleratia if

• Change in velocity

over time.• Either hitting the gas

or hitting the break counts as acceleration.

• Units are m/s2

delta.• Means “change in”

and is calculated by subtracting the initial value from the final value. atvv if

Signs

• In order to differentiate between directions, we will use different signs.

• In general, it doesn’t matter which direction is positive and which is negative as long as they are consistent. However

• Most of the time people make right positive and left negative. Similarly, people usually make up positive and down negative.

• If velocity and acceleration have the same sign, the object is speeding up. If they have opposite signs, the object is slowing down.

Using linear motion equations

• We always assume that acceleration is constant.

• We use vector quantities, not scalar quantities.• We always use instantaneous velocities, not

average velocities (unless specifically stated)• Direction of a vector is indicated by sign.

Incorrect use of signs will result in incorrect answers.

Practice Problem

A car going 15m/s accelerates at 5m/s2 for 3.8s. How fast is it going at the end of the acceleration?

First step is identifying the variables in the equation and listing them.

Practice Problem

A car going 15m/s accelerates at 5m/s2 for 3.8s. How fast is it going at the end of the acceleration?

t=3.8s

vi=15m/s

a=5m/s2

vf=?

Practice Problem 2

• A penguin slides down a glacier starting from rest, and accelerates at a rate of 7.6m/s2. If it reaches the bottom of the hill going 15m/s, how long does it take to get to the bottom?

Practice Problem 2

• A penguin slides down a glacier starting from rest, and accelerates at a rate of 7.6m/s2. If it reaches the bottom of the hill going 15m/s, how long does it take to get to the bottom?

Equation for displacement

t

xv

fi vvv 21

tvd

tvvx fi 21

tvvxx fiif 21

ifif xtvvx 21

Practice Problem

• A cyclist speeds up from his 8.45m/s pace. As he accelerates, he goes 325m in 30s. What is his final velocity?

• A car slows from 45 km/hr to 30km/hr over 6.2s. How far does it travel in that time?

Equation that doesn’t require vf

ifif xtvvx 21 atvv if

iiif xtatvvx 21

iif xatvtx )2(21

iif xattvx 2

21

iif xtvatx 2

21

Practice Problems

• If a car decelerates at a rate of –4.64m/s2

and it travels 162m in 3s, how fast was it going initially?

• A car cruises down the highway at a constant rate of 12.5m/s. A cop pulls out from rest 25m behind the car and accelerates at a rate of 2m/s2. How long will it take the cop to catch up to the speeding car? How fast will the cop be going when he catches the car?

An equation not needing t ifif xtvvx 2

1atvv if

atvv if

ta

vv if i

iffif x

a

vvvvx

2

1

a

vvxx ifif

22

21

222 ifif vvxxa

ifif xxavv 222

A bowling ball is thrown at a speed of

6.8m/s. By the time it hits the pins 63m

away, it is going 5.2m/s. What is the

acceleration?

The Big 4

atvv if ifif xxavv 222

iif xtvatx 2

21 ifif xtvvx 2

1

Gravity• Gravity causes an acceleration.• All objects have the same acceleration due

to gravity.• Differences in falling speed/acceleration

are due to air resistance, not differences in gravity.

• g=-9.8m/s2 (what does the sign mean?)• When analyzing a falling object, consider

final velocity before the object hits the grounds.

Problem Solving Steps

• Identify givens in a problem and write them down.

• Determine what is being asked for and write down with a questions mark.

• Select an equation that uses the variables (known and unknown) you are dealing with and nothing else.

• Solve the selected equation for the unknown.• Fill in the known values and solve equation

Hidden Variables

• Objects falling through space can be assumed to accelerate at a rate of –9.8m/s2.

• Starting from rest corresponds to a vi=0

• A change in direction indicates that at some point v=0.

• Dropped objects have no initial velocity.

Practice Problem

• A ball is thrown upward at a speed of 5m/s. How far has it traveled when it reaches the top of its path and how long does it take to get there?

vi=5m/s

vf=0m/s

a=g=-9.8m/s2

d=?

t=?

An onion falls off an 84m high cliff. How

long does it take him to hit the ground?

An onion is thrown off of the same cliff at

9.5m/s straight up. How long does it take

him to hit the ground?

A train engineer notices a cow on the

track when he is going 40.7m/s. If he can

decelerate at a rate of -1.4m/s2 and the

cow is 500m away, will he be able to stop

in time to avoid hitting the cow?

• A wind up car starts at rest and accelerates at a rate of 0.30m/s2 for 5s before it begins to slow down. At that point, it decelerates at a rate of 0.50m/s2. How far does the car go?

Displacement (Position) vs. Time Graphs

• Position, or displacement can be determined simply by reading the graph.

• Velocity is determined by the slope of the graph (slope equation will give units of m/s).

• If looking for a slope at a specific point (i.e. 4s) determine the slope of the entire line pointing in the same direction. That will be the same as the slope of a specific point.

• What is the position of the object at 7s?

• What is the displacement of the object from 3s to 6s?

• What is the velocity at 2s?

Velocity vs. Time Graphs

• Velocity is determined by reading the graph.

• Acceleration is determined by reading the slope of the graph (slope equation will give units of m/s2).

Velocity vs. Time Graphs• Displacement is found using

area between the curve and the x axis. This area is referred to as the area under the curve (finding area will yield units of m).

• Areas above the x axis are considered positive. Those underneath the x axis are considered negative.

• Break areas into triangles (A=1/2bh), rectangles (A=bh), and trapezoids (A=1/2[b1+ b2]h).

Velocity vs. Time Graphs

• What is the acceleration of the object at 6s?

• What is the displacement of the object at 4s?

• What is the displacement of the object from 3s to 12s?

• What is the velocity of the object at 6s?

• What is the acceleration of the object at 4s?

• What is the displacement of the object at 7s?

• What is the displacement of the object at 10s?

Homework

• Questions– 1, 3-6, 9, 30-36, 46-48, 50, 71-76

• Problems– 10, 11, 14, 16, 28, 37, 40, 45, 53, 55, 57, 59, 64,

70, 77, 80, 86-88, 90, 93, 94, 96, 97, 99, 108

Graph Practice Sheet