Physics 2A. Lecture 2A. "You must learn from the mistakes of others. You can't possibly live long enough to make them all yourself.

Transcription

1 Physics 2A Lecture 2A "You must learn from the mistakes of others. You can't possibly live long enough to make them all yourself." --Sam Levenson 1

2 Motion Chapter 2 will focus on motion in one dimension. Any description of motion involves three concepts: 1) Displacement 2) Velocity 3) Acceleration 2

3 Displacement Nearly the first thing you must do in every Physics problem is define a coordinate system. Choosing the proper coordinate system can make a huge difference. For one dimensional motion it is rather easy. 1) Choose an origin. 2) Choose a positive direction. 3

4 Displacement x is defined as the position compared to the origin. Displacement, x, is a difference between positions. x = x 2 x 1 Distance, d, is the total length travelled. d can only be a positive value. 4

5 2-D Variables If we try to describe motion in two dimensions we turn to the vector, r. Position will now be given by: Displacement will be given by: Note that position depends on where you choose the origin while displacement does not. 5

6 Velocity The velocity of an object is its displacement over a period of time. Average velocity, v avg, is: The speed of an object is its distance travelled over a period of time. Average speed is: Average speed is only a positive value. 6

7 Velocity Graphically, we find average velocity by examining the rise (Δx) over the run (Δt) in an x vs. t graph Between points A and B, we find that the average velocity would be: 7

8 Velocity Instantaneous velocity is the velocity at a given instant of time. Instantaneous velocity, v, is: Instantaneous speed will just be the magnitude of the instantaneous velocity (only positive). We usually drop instantaneous when talking about velocity or speed. 8

9 Velocity Graphically, we find instantaneous velocity by examining the slope of an x vs. t graph at a particular point. It is different from average velocity in that you do not care what happens over a time period only a time instant. 9

10 Velocity Example Runner A is initially 4.0 miles west of a flagpole and is running with a constant velocity of 6.0 mph due east. Runner B is initially 3.0 miles east of the flagpole and is running with a constant velocity of 5.0 mph due west. How far are the runners from the flagpole when they meet? Answer First, you must define a coordinate system. Let s choose the flagpole as the origin with east being the positive x-direction. 10

11 Velocity Answer Each runner needs their own separate variables measured from the flagpole: For Runner A => voa = Δx A t Similarly for Runner B => But recall that displacement is: Δx A = ( v oa )t Δx B = ( v ob )t Δx B = x B, final x B,initial We can thus rewrite our equations for Runner A and B as: x A, final = x A,initial + ( v oa )t x B, final = x B,initial + ( v ob )t 11

12 Velocity Answer When the runners meet their positions will be equal: x B, final = x A, final x B,initial + ( v ob )t = x A,initial + ( v oa )t We know every variable here except for time, so we can plug them in and solve for time. 3.0 miles + ( 5.0 mi hr )t = 4.0 miles + ( 6.0 mi hr )t 7.0 miles = ( 11.0 mi hr)t t = hr = hr <= Yeah we re done! Nope! 12

13 Velocity Answer We wanted to know how far from the flagpole the runners meet. We can use either position equation to solve: x B, final = x B,initial + ( v ob )t = 3.0 miles + ( 5.0 mi ) hr ( hr) x B, final = 0.18 miles where the negative sign means that they meet to the west of the flagpole. 13

14 2-D Variables In two dimensions, average velocity is again the total displacement over the time interval: Since Δt is a scalar and always positive, average velocity will always point in the direction of displacement. Instantaneous velocity is again the velocity of an object at any instant of time. Δ r Δt 0 Δt = d r dt v = lim 14

15 Acceleration The acceleration of an object is how much its velocity changes over a period of time. Average acceleration, a avg, is: Instantaneous acceleration is the acceleration of an object at a given instant of time. Acceleration is: Acceleration is usually measured in units of m/s 2. But you can also find it in other units so be careful. 15

16 Acceleration Watch out for the sign of acceleration. Positive acceleration does not always mean speeding up. Nor does negative acceleration always mean slowing down. When acceleration and velocity are in the same direction, the object speeds up. When acceleration and velocity are in opposite directions, the object slows down. 16

17 2-D Variables In two dimensions, average acceleration is again the change in velocity over the time interval: Since Δt is a scalar and always positive, average acceleration will always point in the direction of the change of velocity. Instantaneous acceleration is again the acceleration of an object at any instant of time. Δv a = lim Δt 0 Δt = d v dt 17

18 For Next Time (FNT) Keep working on the homework for Chapter 2. 18