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Kinematics In One Dimension What Is Kinematics? • Kinematics are the concepts that describe motion. • We don’t care why an object moves. We only care that it moves. • We will be using 5 different quantities to describe the motion of an object. Five Quantities Describing Motion • Displacement (d) – measured in meters • Initial Velocity (vo) – measured in m/s • Final Velocity (vf) – measured in m/s • Acceleration (a) – measured in m/s2 • Time (t) – measured in seconds Positive and Negative • You must assign a positive and a negative direction to all problems. • The positive and negative directions give your values a positive or negative sign and vice versa Displacement • Displacement (x) is a vector that points from an object’s initial position to its final position and has a magnitude that equals the shortest distance between the two positions. Velocity • Velocity is a vector quantity that is given by an object’s change in displacement divided by the time interval during which the change took place. Constant Velocity • When an object is moving at a constant velocity (NO speed changes or direction changes) its velocity can be found by: • v = d/t • d = displacement • t = time How Far Does He Run? • How far does a jogger run in 1.5 hours if his average speed is 2.22 m/s? • 11988 m They’re Good • Ryan Hall runs a 10.0 km course with an average speed of 4.39 m/s. Deena Kastor covers the same distance with an average speed of 4.27 m/s. How much later (in seconds) should Ryan Hall leave to finish at the exact same time as Deena Kastor? Sunday Drive • You drive a car 2.0 hours at 40 km/h, then 2.0 h at 60 km/h. • What is your average velocity? • Do you get the same answer if you drive 100 km at each of the two speeds above? Stop That Car • You’re driving down a street at 55 km/h. Suddenly a child runs into the street. If it takes you 0.75 s to react and apply the brakes, how many meters will you have traveled before you begin to slow down? Relative Velocity • Velocity is based off of a reference point. • Consider when you drive down the road in your car. • What’s your velocity compared to a traffic sign? • What’s your velocity compared to your drink in the cup holder? Buggies Collide • Today you will be colliding two constant velocity buggies. • You will need to find the velocity of both of the buggies you intend to use. • You will set each buggy at opposite ends of a track. • You will have to predict where the buggies will collide. • You are being grades on your accuracy only! Be sure you’re correct before you attempt your collision. Car Crash • A driver falls asleep behind the wheel of a car and drifts into the oncoming lane. The sleeping driver is going 16 m/s and an oncoming car is going 20 m/s. If they are 85 m apart when will they collide and where will they collide using the sleeping driver’s start as your reference point. Acceleration • Acceleration is a vector quantity. It is defined as the change in an object’s velocity over the time interval during which the change takes place. Constant Acceleration • There are 4 equations that we can use to fully describe the motion of an object. • These equations only work for Constant Acceleration Ready For Liftoff • Determine the displacement of a plane that uniformly accelerated from 53 m/s to 75 m/s in 12 seconds. This Is Why You Shouldn’t Speed • A race car can be slowed with a constant acceleration of –8 m/s2. How many meters will it take the car to stop if it is initially traveling at 55 m/s? • What if it was initially moving at 110 m/s? Take Off • An airplane must reach a velocity of 71 m/s for takeoff. If the runway is 1.0 km long, what must the constant acceleration be? Target Practice • If a bullet leaves the muzzle of a rifle with a speed of 600 m/s, and the barrel of the rifle is 0.9 m long, what is the acceleration of the bullet while it is in the rifle? • In which one of the following situations does the car have a westward acceleration? • (a) The car travels westward at constant speed. • (b) The car travels eastward and speeds up. • (c) The car travels westward and slows down. • (d) The car travels eastward and slows down. • (e) The car starts from rest and moves toward the east. Darn Truck • As a traffic light turns green, a waiting car starts with a constant acceleration of 5.0 m/s2. At the instant the car begins to accelerate a truck with a constant velocity of 24 m/s passes in the next lane. • How far will the car travel before it overtakes (is side by side again with) the truck? • How fast will the car be traveling when it overtakes the truck? Where Did Those Equations Come From? • The four kinematics equations of constant acceleration. • How did we get them? • What exactly do they mean? You Have To Consider The Graphs • Consider the 1st equation: • vf = vo + at • Now consider a velocity vs. time graph for an object undergoing constant acceleration. V vs. T d = ½ (vo + vf)t • This also comes from a v vs. t graph (Only under constant acceleration) What About The Others? • The other two equations come from combining the previous two equations. • vf = vo + at & d = ½ (vo + vf) t • d = vo + ½ at2 • v f2 = v o2 + 2ad Graphs • Why have graphs? What do they tell us? • We can learn about the motion by looking at both slope and area under the curve of different graphs. • d vs. t, v vs. t, a vs. t Displacement vs. Time • What does a displacement vs. time graph tell us? • The slope of a line on an x vs. t graph tells you about velocity. • The area under the curve tells you nothing. Velocity vs. Time • What does a velocity vs. time graph tell us? • The slope tells us about the acceleration. • The area under the curve tells us about displacement. Acceleration vs. Time • What does an acceleration vs. time graph tell us? • The slope tells us nothing. • The area under the curve tells us velocity. Questions To Ask Yourself • Is the slope a constant? (a straight line with any slope) • Is the slope not a constant? (a curved line of any kind) Two Types Of Motion • We will deal with only two types of motion in this class. (That does not mean that these are the only two types of motion) • Non-accelerated motion • Constant acceleration motion • Here are what their graphs look like. Non-Accelerated Motion Constant Acceleration Motion Buggies & Fan Cars • We are now going to repeat our Buggies Collide lab only now we are going to be using a Buggy and a Fan Car. • Don’t forget the Fan Car is accelerating. Acceleration Due To Gravity • Acceleration due to gravity (g) is the acceleration that all objects falling toward the Earth possess. • On the surface of the Earth g = 9.8 m/s2 toward the Earth. • Whenever an object is in freefall a = g = 9.8m/s2 down Drop A Rock • A student drops a rock from a bridge to the water 12.0 m below. With what speed does the rock strike the water? • A particle travels along a curved path between two points P and Q as shown. The displacement of the particle does not depend on Q P • (a) The location of P • (b) The location of Q • (c) The distance traveled from P to Q • (d) The shortest distance between P & Q • (e) The direction of Q from P Stuntman • The greatest height reported for a jump into an airbag is 99.4 m by stuntman Dan Koko. In 1948 he jumped from rest from the top of the Vegas World Hotel and Casino. He struck the airbag with a speed of 39 m/s (88 mi/h). To assess the effects of air resistance determine how fast he would have been traveling on impact had air resistance been absent. • Ball A is dropped from rest from a window. At the same instant, ball B is thrown downward; and ball C is thrown upward from the same window. Which statement concerning the balls after their release is necessarily true if air resistance is neglected? • (a) At some instant after it is thrown, the acceleration of ball C is zero. • (b) All three balls strike the ground at the same time. • (c) All three balls have the same velocity at any instant. • (d) All three balls have the same acceleration at any instant. • (e) All three balls reach the ground with the same velocity. Heads Up • Assuming that the arrow lands 2 m below where it was shot find the max height of the arrow and how fast it was moving when it landed. • Two objects A and B accelerate from rest with the same constant acceleration. Object A accelerates for twice as much time as object B, however. Which one of the following statements is true concerning these objects at the end of their respective periods of acceleration? • (a) Object A will travel twice as far as object B. • (b) Object A will travel four times as far as object B. • (c) Object A will travel eight times further than object B. • (d) Object A will be moving four times faster than object B. • (e) Object A will be moving eight times faster than object B. Get Wrecked • A wrecking ball is hanging from a crane when suddenly the cable breaks. The time it takes for the ball to fall halfway to the ground is 1.2 seconds. Find the time it takes for the ball to fall from rest all the way to the ground. • In the process of delivering mail, a postal worker walks 161 m, due east from his truck. He then turns around and walks 194 m, due west. What is the worker’s displacement relative to his truck? • (a) 33 m, due west • (b) 194 m, due west • (c) 355 m, due west • (d) 33 m, due east • (e) 252 m, due east How Does It Move? • Describe the velocity and acceleration over each colored section. • An object moving along a straight line is decelerating. Which one of the following statements concerning the object’s acceleration is necessarily true? • (a) The value of the acceleration is positive. • (b) The direction of the acceleration is in the same direction as the displacement. • (c) An object that is decelerating has a negative acceleration. • (d) The direction of the acceleration is in the direction opposite to that of the velocity. • (e) The acceleration changes as the object moves along the line. Look Out Ahead • While you’re in your car traveling down the road at 23 m/s a dog runs into the road 35 m ahead. If you immediately hit the brakes and slow at a rate of 8 m/s will you be able to stop in time to avoid hitting the dog? How far short or beyond the dog do you stop? • Starting from rest, a particle confined to move along a straight line is accelerated at a rate of 5.0 m/s2. Which one of the following statements accurately describes the motion of this particle? • (a) The particle travels 5.0 m during each second. • (b) The particle travels 5.0 m only during the first second. • (c) The speed of the particle increases by 5.0 m/s during each second. • (d) The acceleration of the particle increases by 5.0 m/s2 during each second. • (e) The final speed of the particle will be proportional to the distance that the particle covers.