Chapter 2: 1D Kinematics

Transcription

1 Chapter 2: 1D Kinematics Description of motion involves the relationship between position, displacement, velocity, and acceleration. A fundamental goal of 1D kinematics is to determine x(t) if given initial conditions and the object s acceleration as a function of time, a(t).

2 Velocity The average velocity during a time interval is given by v avg = x t Δx is the displacement. Instantaneous velocity: x v = lim t!0 t = dx dt

3 Clicker Question Which velocity-versus-time graph goes with this position-versus-time graph on the left?

4 Clicker Question

5 Clicker Question Q2.2 This is the x t graph of the motion of a particle. Of the four points P, Q, R, and S, the speed is greatest at A. point P B. point Q C. point R D. point S E. not enough information in the graph to decide

6 Acceleration Acceleration is the rate of change of velocity. v a = lim t!0 t = dv dt

7 Q2.9 Clicker Question This is the v x t graph for an object moving along the x- axis. Which of the following descrip?ons of the mo?on is most accurate? A. The object is slowing down at a decreasing rate. B. The object is slowing down at an increasing rate. C. The object is speeding up at a decreasing rate. D. The object is speeding up at an increasing rate. E. The object s speed is changing at a steady rate.

8 Kinematic Equations: Constant Acceleration Given initial position (x 0 ) and initial velocity (v 0 ), and a constant acceleration a, what is the velocity and position as a function of time? v(t) =v 0 + at v min = v 0 v max = v 0 + at v avg = v max + v min 2 = v at x x 0 = v avg t = v 0 t at2

9 Example A plane landing on the runway touches down with an initial velocity of 270 km/h. Once it touches down, its acceleration is a constant -4.5 m/s 2. How much runway does the plane use in coming to a stop?

10 Kinematic Equations for Constant Acceleration x x 0 = v avg t = v 0 t at2 v(t) =v 0 + at Note: We can combine the two equations into one that does not include time: x x 0 = 1 2a v2 f v 2 0

11 Freefall due to Gravity Near Earth s Surface Ignoring air resistance, the acceleration of all objects (regardless of mass and velocity) is a = g =9.8 m/s 2

12 Q2.4 Clicker Question You toss a ball straight upward, in the posi?ve direc?on. The ball falls freely under the influence of gravity. At the highest point in the ball s mo?on, A. its velocity is zero and its accelera?on is zero. B. its velocity is zero and its accelera?on is posi?ve (upward). C. its velocity is zero and its accelera?on is nega?ve (downward). D. its velocity is posi?ve (upward) and its accelera?on is zero. E. its velocity is posi?ve (upward) and its accelera?on is zero.

13 Freefall due to Gravity Near Earth s Surface Example: Throw a ball upward with an initial velocity of 28 m/s. What is the maximum height of the ball?

14 Clicker Question: You are standing on top of a building, leaning over the edge. You throw two rocks with the same initial speed, one upwards and one downwards. Both rocks eventually hit the sidewalk below. Which statement is true? 1. Both rocks have the same speed right before they hit the sidewalk (although they don t hit at the same time). 2. The rock thrown upward is moving faster right before it hits the sidewalk (compared to the rock thrown downward). 3. The rock thrown upward is moving slower right before it hits the sidewalk.

15 Chalkboard Question: In a 100 meter dash race, Joe accelerates at a constant rate for 4 seconds up to his maximum speed and then holds this speed for an additional 6 seconds, finishing in 10 seconds. What was Joe s maximum speed?

16 Clicker Question A truck traveling at a constant speed approaches a car stopped at a red light. When the truck is 100m from the car, the light turns green and the car immediately begins to accelerate (such that it doesn t get rear- ended by the truck). Which graph below represents this situa?on? A) B) Green x Yellow x car t car t C) truck truck D) Blue x car Purple: None of these t truck

17 Chalkboard Question You are on a motorcycle a distance L=10 m behind the front of a truck, moving at a constant speed of 27.8 m/s. An oncoming car, initially d=100 m in front of you, is also driving with the same speed. What is the minimum acceleration of your motorcycle such that you can safely pass the truck?

18 Chalkboard Question You are on a motorcycle a distance L=10 m behind the front of a truck, moving at a constant speed of 27.8 m/s. An oncoming car, initially d=100 m in front of you, is also driving with the same speed. What is the minimum acceleration of your motorcycle such that you can safely pass the truck? a 2 2v 0 2L d L

19 Generalized Kinematic Equations Given initial position (x 0 ) and initial velocity (v 0 ), and given the acceleration (a(t)), what is the position and velocity as a function of time? a(t) = dv dt v(t) =v 0 + t t 0 a(t) dt v(t) = dx dt x(t) =x 0 + t t 0 v(t) dt Always true! Very general

20 Constant Acceleration Revisited Let s re-derive the kinematic equations for constant acceleration: v(t) =v 0 + t t 0 adt= v 0 + a(t t 0 )=v 0 + at (if t 0 = 0) x(t) =x 0 + t t 0 vdt= x 0 + v 0 t at2 (if t 0 = 0)