How does distance differ from displacement?

Transcription

1 July :39 AM Chapter 2 Kinematics in One Dimension Newtonian mechanics: kinematics and dynamics Kinematics: mathematical description of motion (Ch 2, Ch 3) Dynamics: how forces affect motion (Ch 4) Summary of Key Concepts position, displacement, distance (2.1) velocity, speed, average velocity, and average speed (2.2) acceleration and average acceleration (2.3) kinematics equations for constant acceleration (2.4, 2.5) freely falling objects (2.6) graphical analysis of one-dimensional kinematics (2.7) What is displacement? Example: An object moves along a straight line with initial position 8 m and final position 3 m. Determine the displacement. How does distance differ from displacement? Example: An object moves along a straight line from position 8 m to position 3 m. Then the object moves along the same line another 2 m to the right. (a) Determine the total displacement. Ch2 Page 1

2 (b) Determine the total distance travelled. How are average velocity and average speed calculated? Example: An object moves along a straight line from position 8 m to position 3 m in 4 s. Then the object moves along the same line another 2 m to the right in 1 s. (a) Determine the average velocity for each leg of the trip, and for the whole trip. (b) Determine the average speed for each leg of the trip, and for the whole trip. Example: Alice jogs one complete lap around an oval track, a distance of 200 m, in a time of 40 s. (a) Determine Alice's displacement and average velocity. (b) Determine Alice's distance travelled and her average speed. Ch2 Page 2

3 Example: Basil drives from St. Catharines to Toronto, a distance of 110 km, and then back again. His average speed on the way to Toronto is 60 km/h, and his average speed on the way back is 90 km/h. Determine Basil's average speed for the whole trip. Question: Does the distance between the two cities matter in the previous example? Can you solve the problem if the distance between the two cities is not given? What is acceleration? How is it calculated? Ch2 Page 3

4 Example: Alice drives at a speed of 50 km/h, then enters a highway, and so increases her speed to 100 km/h in a time of 8 s. Determine her acceleration, in both (km/h)/s and in m/s 2. Example: Alice now leaves the highway, and so decreases her speed from 100 km/h to 60 km/h in a time of 10 s. Determine her acceleration, in both (km/h)/s and in m/s 2. Is acceleration a vector or a scalar? Page 53, Problem 19 The initial velocity and acceleration of four moving objects at a given instant in time are given in the following table. Determine the final speed of each of the objects, assuming that the time elapsed since t = 0 s is 2.0 s. Initial Acceleration Velocity v 0 a Ch2 Page 4

5 (a) +12 m/s +3.0 m/s 2 (b) +12 m/s 3.0 m/s 2 (c.) 12 m/s +3.0 m/s 2 (d) 12 m/s 3.0 m/s 2 Page 53, Problem 23 Two motorcycles are travelling east with different speeds. However, four seconds later, they have the same speed. During this four-second interval, cycle A has an average acceleration of 2.0 m/s 2 east, while cycle B has an average acceleration of 4.0 m/s 2 east. By how much did the speeds differ at the beginning of the four-second interval, and which motorcycle was moving faster? What are the kinematics equations for constant acceleration? How do you use them? Ch2 Page 5

6 Page 53, Problem 24 A basketball player starts from rest and sprints to a speed of 6.0 m/s in 1.5 s. Assuming that the player accelerates uniformly, determine the distance she runs. Page 53, Problem 32 Two rockets are flying in the same direction and are side by side at the instant their retrorockets fire. Rocket A has an initial velocity of m/s, whereas rocket B has an initial velocity of m/s. After a time t both rockets are again side by side, the displacement of each being zero. The acceleration of rocket A is 15 m/s 2. Determine the acceleration of rocket B. Ch2 Page 6

7 Page 54, Problem 34 A race driver has made a pit stop to refuel. After refuelling, he starts from rest and leaves he pit area with an acceleration of magnitude 6 m/s 2 ; after 4.0 s, he enters the main speedway. At the same instant, another car on the speedway travelling at a constant speed of 70.0 m/s overtakes and passes the entering car. The entering car maintains its acceleration. Determine the time needed for the entering car to catch the other car. Page 54, Problem 40 An airplane has a length of 59.7 m. The runway on which the plane lands intersects another runway. The width of the intersection is 25.0 m. The plane decelerates through the intersection at a rate of 5.70 m/s 2 and clears it with a final speed of 45.0 m/s. Determine the time needed for the plane to clear the intersection. Ch2 Page 7

8 Page 54, Problem 42 A train is 92 m long and starts from rest with a constant acceleration at time t = 0 s. At this instant, a car just reaches the end of the train. The car is moving with a constant speed. At a time t = 14 s, the car just reaches the front of the train. Ultimately, however, the train pulls ahead of the car, and at time t = 28 s, the car is again at the rear of the train. Determine the speed of the car and the acceleration of the train. What is free fall? Page 54, Problem 44 A dynamite blast at a quarry launches a chunk of rock straight upward, and 2.0 s later it is rising at a speed of 15 m/s. Assuming air resistance does not exist, calculate the speed of the rock (a) at launch, and (b) 5.0 s after launch. Ch2 Page 8

9 Page 54, Problem 46 A ball is thrown vertically upward, and 8.0 s later the ball returns to its point of release. Determine the ball's initial speed. Page 54, Problem 47 Two identical pellet guns are fired simultaneously from the edge of a cliff. These guns impart an initial speed of 30.0 m/s to each pellet. Gun A is fired straight upward, with the pellet going up and then falling back down, eventually hitting the ground beneath the cliff. Gun B is fired straight downward. In the absence of air resistance, how long after pellet B hits the ground does pellet A hit the ground? Ch2 Page 9

10 Page 54, Problem 63 While standing on a bridge 15.0 m above the ground, you drop a stone from rest. When the stone has fallen 3.20 m, you throw a second stone straight down. What initial speed must you give the second stone if they are both to reach the ground at the same instant? Page 54, Problem 64 A roof tile falls from rest from the top of a building. An observer inside the building notices that it takes 0.20 s for the tile to pass her window, which has a height of 1.6 m. Determine the distance between the top of the window and the roof. Ch2 Page 10

11 How can we describe motion graphically? Position-time graphs for zero acceleration (i.e., constant velocity) Velocity-time graphs for constant, non-zero acceleration Position-time graphs for constant, non-zero acceleration Ch2 Page 11

12 Derivations of the kinematics equations for constant acceleration Page 56, Problem 86 A hot-air balloon is rising straight up at a constant speed of 7.0 m/s. When the balloon is 12.0 m above the ground, a gun fires a pellet straight up from ground level with an initial speed of 30.0 m/s. Along the paths of the balloon and the pellet, there are two places where each of them has the same altitude at the same time. How far above the ground are these places? Ch2 Page 12

13 Page 56, Problem 88 A football player, starting from rest at the line of scrimmage, accelerates along a straight line for a time of 1.5 s. Then, during a negligible amount of time, he changes the magnitude of his acceleration to 1.1 m/s 2. With this acceleration, he continues in the same direction for another 1.2 s, until he reaches a speed of 3.4 m/s. Determine the value of his acceleration (assumed to be constant) during the initial 1.5-s period. Page 56, Problem 81 A woman and her dog are out for a Ch2 Page 13

14 morning run to the river, which is 4.0 km away. The woman runs at 2.5 m/s in a straight line. Her dog is unleashed and runs back and forth at 4.5 m/s between his owner and the river, until the woman reaches the river. Determine the distance travelled by the dog. Ch2 Page 14