Position, speed, velocity, & acceleration. From last time. Quantifying motion: Distance and Time. Can use this information to find the speed

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1 From last time Inertia: tendency of body to continue in straight-line motion at constant speed unless disturbed. Superposition: object responds independently to separate disturbances Galileo used these properties to determine: Light and heavy objects fall identically. Falling time proportional to square of falling distance. But WHY? Gravity disturbs the object, leading to falling motion. But how does this lead to Galileo s results? Mon. Jan Physics 7, Spr6 Position, speed, velocity, & acceleration Need to understand these concepts position distance speed, velocity average instantaneous acceleration average instantaneous Mon. Jan Physics 7, Spr6 A moving object changes its position with time. x = pos. at time t x = pos. at time t Quantifying motion: Distance and Time x, t x, t e.g. at : am, I am meters along the path (x = m, t =: am) at ::5 am, I am 8 meters along the path (x =8 m, t =::5 am) My position at all times completely describes my motion Mon. Jan Physics 7, Spr6 Can use this information to find the speed If I move 5 meters in 5 seconds. Then # meters in each second = 5 divided by 5 = meter per second. e.g. could walk meter in the first second, and meter in the next second, etc. BUT maybe I walked meters in the first second and then 5 meters in 4 seconds. Mon. Jan Physics 7, Spr6 4 The average speed is the same Average speed = As an equation: Distance traveled = d Traveling time = t Average speed = s Could also write d = s t distance traveled traveling time s = d t So knowing average speed lets us find distance traveled Mon. Jan Physics 7, Spr6 5 Instantaneous speed Instantaneous speed is the average velocity over an infinitesimal (very short) time interval. This is what your speedometer reads. Instantaneous speed gives you a better understanding of the motion. Mon. Jan Physics 7, Spr6 6

2 Think about this one: You increase your speed uniformly from to 6 mph. This takes 6. seconds. Your average speed is. Acceleration Acceleration is the rate at which velocity changes: Acceleration = change in velocity time to make the change A. mph B. mph C. 4 mph D. 6 mph Mon. Jan Physics 7, Spr6 7 Mon. Jan Physics 7, Spr6 8 Understanding acceleration Major points Zero acceleration Constant velocity constant acceleration in the same direction as v Increasing velocity constant acceleration opposite of v Decreasing velocity Mon. Jan Physics 7, Spr6 9 position: coordinates of a body velocity: rate of change of position average : change in position instantaneous: average velocity over a very small time interval acceleration: rate of change of velocity average: change in velocity instantaneous: average acceleration over a very small time interval Mon. Jan Physics 7, Spr6 Just to check Why a=? A car s position on a highway is plotted versus time. It turns out to be a straight line. Which of these statements is true? A. Its acceleration is negative B. Its acceleration is positive C. Its acceleration is zero D. Its velocity is zero Position (m) Position vs time is a straight line x = vt Constant velocity Position (m) Zero acceleration Change in position Change in time change in position means constant velocity (= ) Constant velocity means zero acceleration Mon. Jan Physics 7, Spr6 Mon. Jan Physics 7, Spr6

3 What about constant acceleration? Acceleration = Constant acceleration: For every time interval (say, second), the velocity chnages by the same amount. a> gives a uniformly increasing velocity: v = at Constant acceleration change in velocity Velocity (m/s) Constant acceleration Mon. Jan Physics 7, Spr6 Question You are traveling at 6 miles per hour. You apply the brakes, resulting in a constant negative acceleration of - mph / second. How many seconds does it take to stop? A. seconds B. 6 seconds C. seconds Velocity change is mph for every second. Takes six seconds to decrease the velocity to zero Mon. Jan Physics 7, Spr6 4 Questions How far does the car go during that time? A.. mile B.. mile C..5 mile Since speed changes uniformly with time (from 6 mph to mph), so average speed is mph. Distance = average speed x time = ( miles/hour) x (6 seconds) = = ( miles/hour) x (/6 hr) = / mile Mon. Jan Physics 7, Spr6 5 Back to Galileo Use position, velocity, acceleration to quantify the motion of a falling object. Mon. Jan Physics 7, Spr6 6 Distance vs time for falling ball Average speed for falling ball From analyzing the video frame by frame we find the position vs time. This completely describes the motion Distance proportional to time squared Total time=.7s Total distance=.6m Avg speed =.6m/.7s=.5 m/s d = ( 4.9 m /s )t Mon. Jan Physics 7, Spr Mon. Jan Physics 7, Spr6 8

4 .m ".8m speed = = 4.5m /s.488s ".4s.4m ".m speed = =.m /s.75s ".8s Instantaneous speed.6m ".m speed = = 6.5m /s.7s ".68s s.45s.69s Mon. Jan Physics 7, Spr6 9 Instantaneous speed vs time Instantaneous speed proportional to time. So instantaneous speed increases at a constant rate This means constant acceleration s=at change in speed accel = 6.8 m /s =.69 s = 9.85 m /s/s = 9.85 m /s VELOCITY ( m/s) TIME ( s ) Mon. Jan Physics 7, Spr6 Uniform acceleration from rest Falling object: constant acceleration Acceleration = constant = a = 9.8 m/s Velocity = (acceleration)x(time) = at Uniformly increasing velocity Distance = (average vel)x(time) = (/)at x t = (/)at The data show that a falling object has constant acceleration. This is called the acceleration of gravity 9.8 m/s/s = 9.8 m/s But why does gravity result in a constant acceleration? Why is this acceleration independent of mass? Mon. Jan Physics 7, Spr6 Mon. Jan Physics 7, Spr6 Tough questions These are difficult questions. Maybe not completely answered even now. But tied into a more basic question: What causes acceleration? Or, how do we get an object to move? A hot topic in the 7th century. Descartes (cogito ergo sum) was a major player in this. Descartes view Motion and rest are primitive states of a body without need of further explanation. Bodies only change their state when acted upon by an external cause. This is similar our concept of inertia Mon. Jan Physics 7, Spr6 Mon. Jan Physics 7, Spr6 4 4

5 Inertia and momentum Principle of inertia: object continues at constant velocity unless disturbed. So a disturbance will change the velocity. This change in velocity is acceleration. Could start an object moving that is at rest, or stop an object that is moving. An object at rest subject to a disturbance could start moving at some velocity. Different types of objects Objects with lots of inertia don t change motion as much as lighter objects subject to the same disturbance. Inertia measures the degree to which an object at rest will stay at rest. An object with lots of inertia is difficult to accelerate (acceleration = change in velocity). Mon. Jan Physics 7, Spr6 5 Mon. Jan Physics 7, Spr6 demo 6 Momentum Same disturbance applied to different objects results in different velocities (e.g. hitting bowling ball and softball w/hammer). But the product mass velocity is the same (e.g. for the bowling ball and the softball). Momentum = (mass) (velocity) Descartes also said That a body, upon coming in contact with a stronger one, loses none of its motion; but that, upon coming in contact with a weaker one, it loses as much as it transfers to that weaker body So for Descartes, the total amount of motion is always the same. We call the amount of motion momentum, and Descartes law as conservation of momentum Mon. Jan Physics 7, Spr6 7 Mon. Jan Physics 7, Spr6 8 Momentum conservation Can easily describe interactions of objects. The total momentum (sum of momenta of each object) of the system is always the same. We say that momentum is conserved. Momentum can be transferred from one object to the other, but it does not disappear. Descartes was able to move beyond the complicated details of collisions to some basic governing principles. Next time, look at how Newton extended these ideas with his three laws of motion. Builds on Galileo and Descartes, but includes the concept of a force. examples Mon. Jan Physics 7, Spr6 9 Mon. Jan Physics 7, Spr6 5