Linear Motion Chapter 2 Vectors vs Scalars Scalars

Linear Motion Chapter 2

Vectors vs Scalars • Scalars are quantities that have a magnitude, or numeric value which represents a size i. e. 14 m or 76 mph. • Vectors are quantities which have a magnitude and a direction, for instance 12 m to the right or 32 mph east.

Describing how far you’ve gone • Distance d • Scalar • Standard units are meters • A measure of how far you have moved with respect to you (what a pedometer would measure) • Displacement d • 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 12 m. That is the distance traveled. • The person walking starts where she stops, so her displacement is zero.

Measuring how fast you are going • Speed v • Scalar • Standard unit is m/s • Velocity v • Vector • Standard unit is m/s, plus direction

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, d=12 m and t=4 s, so v=3 m/s • For velocity, d=0 and t=4 s, so v=0 m/s

Practice Problem • A boy takes a road trip from Philadelphia to Pittsburgh. The distance between the two cities is 300 km. He travels the first 100 km at a speed of 35 m/s and the last 200 km at 40 m/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 300 km. He travels the first 100 km at a speed of 35 m/s and the last 200 km at 40 m/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 • delta. • Means “change in” and is calculated by subtracting the initial value from the final value. • Change in velocity over time. • Either hitting the gas or hitting the break counts as acceleration. • Units are m/s 2

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 • Direction of a vector is indicated by sign. Incorrect use of signs will result in incorrect answers.

Practice Problem A car going 15 m/s accelerates at 5 m/s 2 for 3. 8 s. 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 15 m/s accelerates at 5 m/s 2 for 3. 8 s. How fast is it going at the end of the acceleration? t=3. 8 s vi=15 m/s a=5 m/s 2 vf=?

Practice Problem 2 • A penguin slides down a glacier starting from rest, and accelerates at a rate of 7. 6 m/s 2. If it reaches the bottom of the hill going 15 m/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. 6 m/s 2. If it reaches the bottom of the hill going 15 m/s, how long does it take to get to the bottom?

Equation for displacement

Practice Problems • A car slows from 45 m/s to 30 m/s over 6. 2 s. How far does it travel in that time? • A cyclist speeds up from his 8. 45 m/s pace. As he accelerates, he goes 325 m in 30 s. What is his final velocity?

Equation that doesn’t require vf

Practice Problems A ball rolling up a hill accelerates at – 5. 6 m/s 2 for 6. 3 s. If it is rolling at 50 m/s initially, how far has it rolled? If a car decelerates at a rate of – 4. 64 m/s 2 and it travels 162 m in 3 s, how fast was it going initially?

An equation not needing t

A bowling ball is thrown at a speed of 6. 8 m/s. By the time it hits the pins 63 m away, it is going 5. 2 m/s. What is the acceleration?

The Big 4

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. 8 m/s 2 • 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. 8 m/s 2. • 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 5 m/s. How far has it traveled when it reaches the top of its path and how long does it take to get there? vi=5 m/s d=? vf=0 m/s t=? a=g=-9. 8 m/s 2

A plane slows on a runway from 207 km/hr to 35 km/hr in about 527 m. a. What is its acceleration? b. How long does it take?

An onion falls off an 84 m high cliff. How long does it take him to hit the ground?

An onion is thrown off of the same cliff at 9. 5 m/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. 7 m/s. If he can decelerate at a rate of -1. 4 m/s 2 and the cow is 500 m away, will he be able to stop in time to avoid hitting the cow?

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. 4 s) 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 velocity of the object at 4 seconds?

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/s 2).

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=1/2 bh), rectangles (A=bh), and trapezoids (A=1/2[b 1+ b 2]h).

Velocity vs. Time Graphs • What is the acceleration of the object at 6 s? • What is the displacement of the object at 4 s? • What is the displacement of the object from 3 s to 12 s?

Homework • Problems Required: 3, 9, 10, 12, 13, 17, 20, 22, 28, 30, 31, 33, 34, 38, 41, 45, 47, 49, 54 Additional: 1, 2, 4 -7, 11, 14 -16, 18, 19, 21, 23, 29, 35, 39, 42 -44, 46, 48, 55 -57, 60 • Graph Practice Sheet

• Graph packet • Graph worksheets from old book • Investigations 1, 2, 3 computer labs