Vectors. Coordinates & Vectors. Chapter 2 One-Dimensional Kinematics. Chapter 2 One-Dimensional Kinematics

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1 Chapter 2 One-Dimensional Kinematics Chapter 2 One-Dimensional Kinematics James Walker, Physics, 2 nd Ed. Prentice Hall One dimensional kinematics refers to motion along a straight line. Even though we live in a 3-dimension world, motion can often be abstracted to a single dimension. Terms we will use: Position, distance, displacement Speed, velocity (average and instantaneous) Acceleration (average and instantaneous) Coordinates & Vectors Vectors Not only do we always use a quantitative measure for distance and time, we also intuitively use vectors and coordinates for describing locations and displacements. For example, give directions from CNU to the Virginia Beach Boardwalk. 2.5 mi SE on Warwick Blvd 25 mi SE on I-64 East 12 mi NE on I-264 East; Each instruction includes both distance and direction To define a position in (2-dim) space, we define an origin and a coordinate system (x,y), (N,E) etc. A point P can be described as an ordered pair (x,y) = (4m,3m) or a vector of length 5 m pointing 36.8 = 37 counter-clockwise from the x- axis. We can think of the point P as either a position, or a displacement: P is 4 m along the x-axis plus 3 m along the y-axis y 37 P x 1

2 Coordinate Systems A coordinate system is used to describe location. Example (-3,5) y A coordinate system consists of: a fixed reference point called the origin a set of axes a definition of the coordinate variables Cartesian (4,2) The position of an object is its location in a coordinate system. x Distance and displacement Distance: is the total length of travel. It is always positive. It is measured by the odometer in your car. Displacement: is defined as the change in position of an object x x f x i (Delta)=change x f = final value of x, x i = initial value of x Change can be positive, negative or zero. Displacement is a vector (see Chapter 3) Example Referring to the figure below, you walk from the park to your friend's house, then back to your house. (a) What is the distance traveled? (b) What is your displacement? If you walk from your house to the library, then back to your house, repeat (a) and (b). 2

3 Average Speed and Velocity Example 1 Speed and velocity are not the same in physics! Speed is rate of change of distance: distance average speed = elaspsed time (always positive, no direction) Velocity is rate of change of displacement: displacement x f xi average velocity = = elaspsed time t t velocity is a vector (see Chapter 3) SI units of speed and velocity are m/s. f i (positive, negative or zero with direction) Here we are just giving the x-component of velocity, assuming the other components are either zero or irrelevant to our present discussion Example 2 Referring to figure below, an athlete runs from A to B and back to A. Neglecting the corner distances (a) What is the distance traveled? (b) What is the displacement? (c) What is average velocity if it took the runner 5 minutes for 1 lap. Repeat for average speed. Position vs. Time Plots The average velocity between two times is the slope of the straight line connecting those two points. average velocity from 0 to 3 sec is positive average velocity from 2 to 3 sec is negative 3

4 Instantaneous Velocity The velocity at one instant in time is known as the instantaneous velocity and is found by taking the average velocity for smaller and smaller time intervals: x v lim t 0 t The speedometer indicates instantaneous velocity ( t 1 s). On an x vs t plot, the slope of the line tangent to the curve at a point in time is the instantaneous velocity at that time. Acceleration Often, velocity is not constant, rather it changes with time. The rate of change of velocity is known as acceleration. v v f vi aav = positive, negative or zero t t f ti This is the average acceleration. Acceleration is a vector (see Chapter 3). The unit of acceleration is: m/s 2 Car acceleration is often described in units miles per hour per sec Acceleration of 0 to 60 miles per hour in 8 sec = (60mi/hr-0mi/hr)/8sec =7.5mi/(hr*s) =? m/s/s Instantaneous Acceleration If we wish to know the instantaneous acceleration, we once again let t 0: a lim t 0 v t Velocity vs. Time Plots Graphically, acceleration can be found from the slope of a velocity vs. time curve. For these curves, the average acceleration and the instantaneous acceleration are the same, because the acceleration is constant. 4

5 Deceleration refers to decreasing speed Deceleration is not the same as negative acceleration occurs when velocity and acceleration have opposite signs Example: A ball thrown up in the air. The velocity is upward but the acceleration is downward. The ball is slowing down as it moves upward. (Once the ball reaches its highest point and starts to fall again, it is no longer decelerating.) Example: Position vs. Time Plot (a) Without performing a calculation, indicate whether the velocity is positive, negative, or zero during the segments of the graph labeled A, B, C, and D. (b) Calculate the average velocity for each segment If up is our convention for positive, then both when the ball is rising and falling, the acceleration is negative Example be careful with signs A car moves from a position of +4 m to a position of 1 m in 2 seconds. The initial velocity of the car is 4 m/s and the final velocity is 1 m/s. (a) What is the displacement of the car? (b) What is the average velocity of the car? (c) What is the average acceleration of the car? Motion with Constant Acceleration If acceleration is constant, there are four useful formulae relating position x, velocity v, acceleration a at time t: v = v + at v 0 0 = initial velocity x = x + 1 ( v + v) t x 0 = initial position t x = x + v t = initial time assumed 0 0 at 2 here to be at 0 s v = v0 + 2a( x x0) = v0 + 2a x (Instead of x f, x i, we are using x and x 0) If t 0 0, replace t in these formulae with t t 0 5

6 Note that we are applying restrictions and defining variables. BE CAREFUL WHEN USING A FORMULA! Where do these formulae come from? If acceleration is constant, then a = average acceleration. a=(v-v 0) / (t 0) (a) (t) = (v-v 0) 1): v= v 0 + a t If a = constant, then velocity vs time graph is a straight line For a straight line graph: v ave = (v+v 0)/2 v But v ave=(x-x 0)/(t-0) v 0 (x-x 0)/(t-0) = (v+v 0)/2 t 2): x = x 0 + (v+v 0)t/2 Subsitute 1) into 2) 3): x = x 0 + v 0 t + a t 2 /2 When a chameleon captures an insect, its tongue extends 12 cm in 0.08 s. (a) Find the acceleration of the chameleon's tongue, assuming it to be constant. (b) Let x be the amount that the chameleon's tongue moves during the first 0.040, calculate x. A 27 pound meteorite struck a car, leaving a dent 22 cm deep in the trunk. If the meteorite struck the car with a speed of 590 m/s, what was the magnitude of its deceleration, assuming it to be constant? 6

7 Freely Falling Objects Near the earth s surface, the acceleration due to gravity g is roughly constant: g = a Earth s surface = 9.81 m/s 2 toward the center of the earth Graphical example: A ball is thrown upward from the ground level. Free fall is the motion of an object subject only to the influence of gravity (not air resistance). An object is in free fall as soon as it is released, whether it is dropped from rest, thrown downward, or thrown upward Question: What about the mass of an object? Answer: The acceleration of gravity is the same for all objects near the surface of the Earth, regardless of mass. x = ball s height above the ground Why is acceleration negative? Is there ever deceleration? velocity is positive when the ball is moving upward 1. On a hot summer day several swimmers decide to dive from a railroad bridge into the river below. The swimmers step off the bridge and hit the water approximately 1.5 s later. 2. An astronaut on Saturn drops a rock straight downward from a height of 0.80 m. If the acceleration of gravity on Saturn is 10.5 m/s 2, what is the speed of the rock when it lands?. 7

8 Problem Solving Strategy Make a list of given quantities Make a sketch Draw coordinate axes identify the positive direction Identify what is to be determined Be consistent with units Check that the answer seems reasonable Remain calm While riding on an elevator descending with a constant speed of 3.0 m/s, you accidentally drop a book from under your arm. (a) How long does it take for the book to reach the elevator floor, 1.2 m below your arm? (b) What is the book s speed when it hits the elevator floor? A model rocket rises with constant acceleration to a height of 3.2 m, at which point its speed is 26.0 m/s. (a) How much time does it take for the rocket to reach this height? (b) What was the rocket's acceleration? (c) Find the height and speed of the rocket 0.10 s after launch. 8