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Linear Motion

Learn the difference between scalars and vectors in linear motion. Explore concepts such as distance, displacement, speed, velocity, and acceleration. Understand how to interpret graphs and solve problems using linear motion equations.

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Linear Motion

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  1. Linear Motion Chapter 2

  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.

  3. 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). Describing how far you’ve gone

  4. 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.

  5. Distance vs. Displacement Distance-Add all the distances together, totals 13m. Start 6m Displacement-Measured from beginning to end. 3m Add the left/right pieces and the up/down pieces and use the Pythagorean Theorem. 3m 1m End

  6. Start 6m 3m 3m 1m End Distance vs. Displacement 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.

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

  8. If it take the person 4 seconds to walk around the rectangle, what is her average speed and average velocity? For speed, d=12m and t=4s, so v=3m/s For velocity, d=0 and t=4s, so v=0m/s Velocity and Speed

  9. 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?

  10. 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?

  11. 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 your car reads. Can change from moment to moment. Different types of velocity and speed

  12. Displacement (Position) vs. Time Graphs • 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? • 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.

  13. Acceleration • Change in velocity over time. • Either hitting the gas or hitting the brake counts as acceleration. • Units are m/s2 • delta. • Means “change in” and is calculated by subtracting the initial value from the final value.

  14. 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.

  15. 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).

  16. 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).

  17. 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?

  18. 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?

  19. 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.

  20. 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.

  21. 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=?

  22. 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?

  23. 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?

  24. Equation for displacement

  25. Practice Problems • A car slows from 45 m/s to 30m/s over 6.2s. How far does it travel in that time? • A cyclist speeds up from his 8.45m/s pace. As he accelerates, he goes 325m in 30s. What is his final velocity?

  26. Equation that doesn’t require vf

  27. A ball rolling up a hill accelerates at –5.6m/s2 for 6.3s. If it is rolling at 50m/s initially, how far has it rolled? If a car decelerates at a rate of –4.64m/s2 and it travels 162m in 3s, how fast was it going initially? Practice Problems

  28. An equation not needing t

  29. 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?

  30. The Big 4

  31. 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.

  32. 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

  33. 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.

  34. 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=?

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

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

  37. 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?

  38. 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?

  39. 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

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

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