©2008 by W.H. Freeman and Company Chapter 2 Motionin One Dimension

©2008 by W.H. Freeman and Company 2-1 Displacement There is 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.

©2008 by W.H. Freeman and Company 2-1 Displacement Left: Displacement is positive. Right: Displacement is negative. The displacement is written:

©2008 by W.H. Freeman and Company Average Velocity Speed: how far an object travels in a given time interval Velocity includes directional information:

©2008 by W.H. Freeman and Company A particle at t1 = -2.0 s is at x1= 3.4 cm and at t2 = 4.5 s is at x2= 8.5 cm. What is its average velocity? Can you calculate its average speed from these data? The average velocity is given by The average speed cannot be calculated. To calculate the average speed, we would need to know the actual distance traveled, and it is not given.

©2008 by W.H. Freeman and Company An airplane travels 3100 km at a speed of 790 km/h and then encounters a tailwind that boosts its speed to 990 km/h for the next 2800 km. What was the total time for the trip? What was the average speed of the plane for this trip? The average speed of the plane for the entire trip is. The average speed for each segment of the trip is given by, so for each segment. For the second segment, Thus the total time is For the first segment,

©2008 by W.H. Freeman and Company Acceleration Acceleration is the rate of change of velocity.

©2008 by W.H. Freeman and Company Acceleration Acceleration is a vector, although in one- dimensional motion we only need the sign. The previous image shows positive acceleration; here is negative acceleration:

©2008 by W.H. Freeman and Company 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.

©2008 by W.H. Freeman and Company Motion with Constant Acceleration The average velocity of an object during a time interval t is The acceleration, assumed constant, is

©2008 by W.H. Freeman and Company 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:

©2008 by W.H. Freeman and Company Motion with 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.

©2008 by W.H. Freeman and Company At highway speeds, a particular automobile is capable of an acceleration of about 1.6 m/s 2. At this rate, how long does it take to accelerate from 80 km/h to 110 km/h?. The time can be found from the average acceleration,

©2008 by W.H. Freeman and Company A world-class sprinter can burst out of the blocks to essentially top speed (of about 11.5 m/s) in the first 15.0 m of the race. What is the average acceleration of this sprinter, and how long does it take her to reach that speed? The sprinter starts from rest. The average acceleration is found from the elapsed time is found by solving

©2008 by W.H. Freeman and Company A car slows down uniformly from a speed of 21.0 m/s to rest in 6.00 s. How far did it travel in that time?. The words “slowing down uniformly” implies that the car has a constant acceleration. The distance of travel is found form

©2008 by W.H. Freeman and Company

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.

©2008 by W.H. Freeman and Company 2-7 Falling Objects In the absence of air resistance, all objects fall with the same acceleration, although this may be hard to tell by testing in an environment where there is air resistance.

©2008 by W.H. Freeman and Company

2-7 Falling Objects The acceleration due to gravity at the Earth’s surface is approximately 9.80 m/s 2.

©2008 by W.H. Freeman and Company A baseball is hit nearly straight up into the air with a speed of 22 m/s. (a) How high does it go? (b) How long is it in the air? (a) The displacement can be found from Eq. 2-11c, with x replaced by y. (b)The time of flight can be found from Eq. 2-11b, with x replaced by y, using a displacement of 0 for the displacement of the ball returning to the height from which it was hit. The result of t = 0 s is the time for the original displacement of zero (when the ball was hit), and the result of t = 4.5 s is the time to return to the original displacement. Thus the answer is t = 4.5 seconds. to be at the height where the ball was hit. Choose upward to be the positive direction, and take at the top of the path, andFor the upward path,

©2008 by W.H. Freeman and Company Integration Integration is related to the area under a curve. It is also the inverse of differentiation The total area under the curve from t 1 to t 2 is

©2008 by W.H. Freeman and Company Integration

©2008 by W.H. Freeman and Company Average Velocity

©2008 by W.H. Freeman and Company Instantaneous Velocity The instantaneous velocity is the average velocity, in the limit as the time interval becomes infinitesimally short.

©2008 by W.H. Freeman and Company Acceleration The instantaneous acceleration is the average acceleration, in the limit as the time interval becomes infinitesimally short.