Chapter 2: Describing Motion: Kinematics in One Dimension

1. Chapter 2: Describing Motion: Kinematics in One Dimension Topics we will cover in this chapter: • Reference frames • Displacement • Velocity – average and instantaneous • Acceleration • Constant acceleration (a special case) • How to solve kinematics problems • Free fall (constant acceleration with gravity)

2. 2-1 Reference Frames and Displacement Any measurement of position, distance, or speed must be made with respect to a reference frame. For example, if you are sitting on a train and someone walks down the aisle, the person’s speed with respect to the train is a few miles per hour, at most. The person’s speed with respect to the ground is much higher.

3. 2-1 Reference Frames and Displacement We make a distinction between distance and displacement. Displacement (blue line) is how far the object is from its starting point, regardless of how it got there. Distance traveled (dashed line) is measured along the actual path.

4. 2-1 Reference Frames and Displacement The displacement is written: Δ is the Greek letter delta. Right: Displacement is negative. Left: Displacement is positive.

5. 2-2 Average Velocity Speed is how far an object travels in a given time interval: Velocity includes directional information:

6. 2-2 Average Velocity Example (Problem 7): A horse canters away from its trainer in a straight line, moving 116 m away in 14.0 s. It then turns abruptly and gallops halfway back in 4.8 s. Calculate its average speed and its average velocity for the entire trip, using “away from the trainer” as the positive direction.

7. 2-3 Instantaneous Velocity The instantaneous velocity is the average velocity in the limit as the time interval becomes infinitesimally short. Ideally, a speedometer would measure instantaneous velocity; in fact, it measures average velocity, but over a very short time interval.

8. 2-3 Instantaneous Velocity The instantaneous speed always equals the magnitude of the instantaneous velocity; it only equals the average velocity if the velocity is constant.

9. 2-3 Instantaneous Velocity On a graph of a particle’s position vs. time, the instantaneous velocity is the tangent to the curve at any point.

10. 2-3 Instantaneous Velocity Example (Problem 9): The position of a rabbit along a straight tunnel as a function of time is plotted below. What is its instantaneous velocity at t = 10.0 s? What is its average velocity between t = 40.0 s and t = 50.0 s?

11. 2-4 Acceleration Acceleration is the rate of change of velocity. Example (Problem 20): A sports car accelerates from rest to 95 km/h in 4.5 s. What is its average acceleration in m/s2?

12. 2-4 Acceleration There is a difference between negative acceleration and deceleration: Negative acceleration is acceleration in the negative direction as defined by the coordinate system. Deceleration occurs when the acceleration is opposite in direction to the velocity.

13. 2-4 Acceleration Exercise: A car moves along the x axis. What is the sign of the car’s acceleration if it is moving in the positive x direction with: • Increasing speed? • Decreasing speed? What is the sign of the acceleration if the car moves in the negative x direction with: c. Increasing speed? d. Decreasing speed?

14. 2-4 Acceleration The instantaneous acceleration is the average acceleration in the limit as the time interval becomes infinitesimally short.

15. 2-4 Acceleration Example (Problem 27): A particle moves along the x axis. Its position as a function of time is given by x = 6.8t + 8.5t2, where t is in seconds and x is in meters. What is the acceleration as a function of time?

16. 2-5 Motion at Constant Acceleration The average velocity of an object during a time interval t is The acceleration, assumed constant, is

17. 2-5 Motion at Constant Acceleration In addition, as the velocity is increasing at a constant rate, we know that Combining these last three equations, we find:

18. 2-5 Motion at Constant Acceleration We can also combine these equations so as to eliminate t: We now have all the equations we need to solve constant-acceleration problems.

19. 2-6 Solving Problems • Read the whole problem and make sure you understand it. Then read it again. • Decide on the objects under study and what the time interval is. • Draw a diagram and choose coordinate axes. • Write down the known (given) quantities, and then the unknown ones that you need to find. • What physics applies here? Plan an approach to a solution.

20. 2-6 Solving Problems 6. Which equations relate the known and unknown quantities? Are they valid in this situation? Solve algebraically for the unknown quantities, and check that your result is sensible (correct dimensions). 7. Calculate the solution and round it to the appropriate number of significant figures. 8. Look at the result—is it reasonable? Does it agree with a rough estimate? 9. Check the units again.

21. 2-6 Solving Problems Note that the part of this process that involves looking up the right equation will work fine for kinematics problems (which is what we are doing in this chapter and the next). Once we get to dynamics, we will modify the way we go about solving problems.

22. 2-6 Solving Problems Example (Problem 38): In coming to a stop, a car leaves skid marks 85 m long on the highway. Assuming a deceleration of 4.00 m/s2, what was the speed of the car just before braking?

23. 2-7 Freely Falling Objects Near the surface of the Earth, all objects experience approximately the same acceleration due to gravity. This is one of the most common examples of motion with constant acceleration.

24. 2-7 Freely Falling Objects In the absence of air resistance, all objects fall with the same acceleration, although this may be tricky to tell by testing in an environment where there is air resistance. Dropping a hammer and a feather on the moon

25. 2-7 Freely Falling Objects The acceleration due to gravity at the Earth’s surface, which we call g, is approximately 9.80 m/s2. At a given location on the Earth and in the absence of air resistance, all objects fall with the same constant acceleration. To solve problems involving free fall, use the equations for constant acceleration with a = -9.80 m/s2.

26. 2-7 Freely Falling Objects Example (Problem 51): A baseball is hit almost straight up into the air with a speed of about 20 m/s. • How high does it go? • How long is it in the air?