QuickCheck. A cart slows down while moving away from the origin. What do the position and velocity graphs look like? Slide 2-65

Transcription

1 QuickCheck A cart slows down while moving away from the origin. What do the position and velocity graphs look like? Slide 2-65

2 QuickCheck A cart speeds up toward the origin. What do the position and velocity graphs look like? Slide 2-67

3 Motion with Constant Acceleration The SI units of acceleration are (m/s)/s, or m/s 2. It is the rate of change of velocity and measures how quickly or slowly an object s velocity changes. The average acceleration during a time interval t is: Graphically, a avg is the slope of a straight-line velocityversus-time graph. Acceleration, like velocity, is a vector quantity and has both magnitude and direction. Slide 2-64

4 Example Slide 2-71

5 Example Slide 2-73

6 QuickCheck A cart slows down while moving away from the origin. What do the velocity and acceleration graphs look like? Slide 2-77

7 QuickCheck A cart speeds up while moving toward the origin. What do the velocity and acceleration graphs look like? Slide 2-79

8 Which velocity-versus-time graph or graphs goes with this acceleration-versus-time graph? The particle is initially moving to the right.

9 Example 1 The above figure shows the velocity graph of a train that starts from the origin a. Find the acceleration of the train at t = 3.0 s b. Draw an acceleration graph for the train.

10 The Kinematic Equations of Constant Acceleration Suppose we know an object s velocity to be v is at an initial time t i. We also know the object has a constant acceleration of a s over the time interval t t f t i. We can then find the object s velocity at the later time t f as: Slide 2-81

11 The Kinematic Equations of Constant Acceleration Suppose we know an object s position to be s i at an initial time t i. It s constant acceleration a s is shown in graph (a). The velocity-versus-time graph is shown in graph (b). The final position s f is s i plus the area under the curve of v s between t i and t f : Slide 2-82

12 The Kinematic Equations of Constant Acceleration Suppose we know an object s velocity to be v is at an initial position s i. We also know the object has a constant acceleration of a s while it travels a total displacement of s s f s i. We can then find the object s velocity at the final position s f : Slide 2-83

13 The Kinematic Equations of Constant Acceleration Slide 2-84

14 The Kinematic Equations of Constant Acceleration Motion with constant velocity and constant acceleration. These graphs assume s i = 0, v is > 0, and (for constant acceleration) a s > 0. Slide 2-85

15 Tactics: Drawing a Pictorial Representation Slide 1-71

16 Tactics: Drawing a Pictorial Representation Slide 1-72

17 Example 2 - A car accelerates at 2.0 m/s 2 along a straight road. It passes two markers that are 30 m apart at times t = 4.0 s and t = 5.0 s. What was the car s velocity at t = 0 s?

18 Example 3 - A driver has a reaction time of 0.50 seconds, and the maximum deceleration of her car is 6.0 m/s 2. She is driving at 20 m/s when suddenly she sees an obstacle in the road 50 meters in front of her. Can she stop the car in time to avoid a collision?

19 Using calculus to find position, velocity, and acceleration

20 Example 4 - A particle moving along the x-axis has its position described by the function x(t) = 2t 2 t + 1 meters, where t is in seconds. At t = 2 s what are the particle s (a) position, (b) velocity, and (c) acceleration? Does the particle have a turning point? If so, at what time? Derivative Power Rule for polynomials:

21 Free Fall The motion of an object moving under the influence of gravity only, and no other forces, is called free fall. Two objects dropped from the same height will, if air resistance can be neglected, hit the ground at the same time and with the same speed. Consequently, any two objects in free fall, regardless of their mass, have the same acceleration: In the absence of air resistance, any two objects fall at the same rate and hit the ground at the same time. The apple and feather seen here are falling in a vacuum. Slide 2-94

22 QuickCheck 2.18 A ball is tossed straight up in the air. At its very highest point, the ball s instantaneous acceleration a y is A. Positive. B. Negative. C. Zero. Slide 2-96

23 Example 5 - A student standing on the ground throws a ball straight up. The ball leaves the student s hand with a speed of 15 m/s when the hand is 2.0 m above the ground. How long is the ball in the air before it hits the ground? Slide 2-98

24 Motion on an Inclined Plane Figure (a) shows the motion diagram of an object sliding down a straight, frictionless inclined plane. Figure (b) shows the the free-fall acceleration the object would have if the incline suddenly vanished. This vector can be broken into two pieces: and. The surface somehow blocks, so the one-dimensional acceleration along the incline is The correct sign depends on the direction the ramp is tilted. Slide 2-102

25 Example 6 Motion on an Inclined Plane. A skier is gliding along at 3.0 m/s on horizontal, frictionless snow. He suddenly starts down a 10º incline. His speed at the bottom is 15 m/s. a) What is the length of the incline? b) How long does it take him to reach the bottom?

26 Example?? Race Kinematics: David is driving a steady 30 m/s when he passes Tina, who is sitting in her car at rest. Tina begins to accelerate at a steady 2.0 m/s 2 at the instant when David passes. a) How far does Tina drive before passing David? b) What is her speed as she passes him?