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.

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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 Distance  length of the path between two points  Δx Scalar Standard units are meters A measure of how far you have moved with respect to you (what a pedometer would measure) Displacement  length of the shortest path between two points  Δx 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 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 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 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, 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 Change in velocity over time. Either hitting the gas or hitting the break counts as acceleration. Units are m/s 2   delta. Means “change in” and is calculated by subtracting the initial value from the final value.

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/s 2 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/s 2 for 3.8s. How fast is it going at the end of the acceleration? t=3.8s v i =15m/s a=5m/s 2 v f =?

Practice Problem 2 A penguin slides down a glacier starting from rest, and accelerates at a rate of 7.6m/s 2. 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/s 2. If it reaches the bottom of the hill going 15m/s, how long does it take to get to the bottom?

Equation for displacement

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 v f

Practice Problems If a car decelerates at a rate of –4.64m/s 2 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/s 2. 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

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

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/s 2 (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/s 2. Starting from rest corresponds to a v i =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? v i =5m/s v f =0m/s a=g=-9.8m/s 2 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/s 2 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/s 2 for 5s before it begins to slow down. At that point, it decelerates at a rate of 0.50m/s 2. 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/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=1/2bh), 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 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

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