Linear Motion Chapter 2. Vectors vs Scalars Scalars are quantities that have a magnitude, or numeric...
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Transcript of Linear Motion Chapter 2. Vectors vs Scalars Scalars are quantities that have a magnitude, or numeric...
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