Day 3 – May 9 – WBL Chapter 2 Kinematics: Description of Motion PC141 Intersession 2013Slide 1 A bit of terminology… Kinematics (the topic of the next two chapters) deals only with the description of motion, without considering its causes. It involves concepts of position, displacement, speed, velocity, and acceleration. Dynamics (the rest of PC141) concerns the causes of motion. It begins with the subject of force, which leads to the concepts of mechanical energy, linear momentum, and angular momentum. These are then linked back to the kinematic concepts in order to fully describe the motion.

Day 3 – May 9 – WBL All motion involves changing an object’s position. In studying kinematics, we wish to analyze (or predict) where an object is (or will be) at various times. The simplest description of motion is the distance that an object travels, which is the total path length traversed in moving between two points. Distance depends not only on the position of the two points, but on the particular path taken during the trip. 2.1 Distance and Speed PC141 Intersession 2013Slide 2 Distance is a scalar quantity (it has a magnitude – and units – but no direction). The text uses the symbol d to represent distance.

Day 3 – May 9 – WBL Distance and Speed PC141 Intersession 2013Slide 3

Day 3 – May 9 – WBL We can also define an instantaneous speed s as the speed at a particular moment in time. That is, it is the average speed in the limit as Δt approaches zero. The speedometer in a car shows instantaneous speed (more or less…all sensors have a finite “sensing time”, but it’s very short). Instantaneous speed is very important in calculus-based physics courses, but we won’t use it very often in PC Distance and Speed PC141 Intersession 2013Slide 4

Day 3 – May 9 – WBL For the remainder of this chapter, we will only consider motion in one dimension – that is, objects are constrained to move back and forth along a straight line. This may seem like a very restrictive view of the world, but it will turn out to be quite useful D Displacement and Velocity PC141 Intersession 2013Slide 5 To begin, we will label the straight line as the x-axis. The position of an object is a function of time, labeled x(t). Its value depends on where we place the origin, where x = 0. This location is entirely up to us. ORIGIN

Day 3 – May 9 – WBL Displacement is the straight-line distance between two points, along with an indication of the direction (which is positive if x increases from the first point to the second, and negative if x decreases). The man in the figure has a displacement of +8 m if he walks from x 1 to x 2, where x 1 = x(t 1 ) and x 2 = x(t 2 ) D Displacement and Velocity PC141 Intersession 2013Slide 6 Mathematically, we write Since it has both a magnitude and a direction, displacement is a vector quantity.

Day 3 – May 9 – WBL D Displacement and Velocity PC141 Intersession 2013Slide 7 Average velocity and average speed are not the same thing. For example, if you take a walk to the corner store and back, you might travel a total of 300 m in 5 minutes. Your speed is (300 m) / (300 s) = 1 m/s. However, since your starting position and ending position are the same, your displacement is Δx = 0, so your average velocity is zero.

Day 3 – May 9 – WBL As with speed, we can also define an instantaneous velocity v as the velocity at a particular moment in time. That is, it is the average velocity in the limit as Δt approaches zero: Uniform motion is motion in which the velocity is constant (in both magnitude and direction). In uniform motion, the average velocity and instantaneous velocity are identical. 2.2 Displacement and Velocity PC141 Intersession 2013Slide 8

Day 3 – May 9 – WBL Displacement and Velocity PC141 Intersession 2013Slide 9 If the velocity is constant, then the slope can never change. In this case, the graph is a straight line.

Day 3 – May 9 – WBL If the velocity is not constant, then the graph is a curve. The plot below shows position vs. time for an object that speeds up, slows down, reverses direction, etc. We will discuss it in class. 2.2 Displacement and Velocity PC141 Intersession 2013Slide 10

Day 3 – May 9 – WBL When an object’s velocity is not constant, we say that the object has a non-zero acceleration. The relationship between acceleration and velocity is identical to that between velocity and position. That is, acceleration is the time rate of change of velocity. The average acceleration is defined as SI units for acceleration are m/s 2. Don’t waste too much time contemplating the concept of a “squared second”… the units merely refer to the fact that the ratio of a velocity to a time is measured as (m/s) / s. We can also express the instantaneous acceleration as 2.3 Acceleration PC141 Intersession 2013Slide 11

Day 3 – May 9 – WBL Since velocity is a vector, so is acceleration. A change in velocity might indicate a change in speed, or a change in direction, or both. Either of these will produce a non-zero acceleration. The case of a change in speed (not direction) is shown below. The directions of velocity and acceleration and their relation to whether an object is “speeding up” or “slowing down” are rather confusing at first. 2.3 Acceleration PC141 Intersession 2013Slide 12

Day 3 – May 9 – WBL Acceleration PC141 Intersession 2013Slide 13

Day 3 – May 9 – WBL The case of a change in direction (not speed) is shown below on the left. We won’t analyze this one quite yet – since this motion takes place in 2 dimensions, we will save it for the next chapter. It is also possible to change both speed and direction, as shown on the right. 2.3 Acceleration PC141 Intersession 2013Slide 14

Day 3 – May 9 – WBL In general, acceleration can be a function of time, a(t). However, when acceleration is constant, we can derive many simple relationships among acceleration, velocity, and position. The text implies that this case is examined “for simplicity”… I disagree! There are many examples of physical situations for which acceleration really is constant. In fact, much of PC141 is concerned with these situations. To begin, we need to adjust our notation a bit. Until now, we assumed that there were two specific times, t 1 and t 2, at which an object had position x 1 and x 2 and velocity v 1 and v 2. Here, we will change the initial conditions to t 0 = 0, x 0, and v 0, and consider the final conditions as variables t, x, and v. Note that we can set the initial time to zero since only changes in time have any physical meaning. 2.3 Acceleration PC141 Intersession 2013Slide 15

Day 3 – May 9 – WBL Acceleration PC141 Intersession 2013Slide 16

Day 3 – May 9 – WBL Acceleration PC141 Intersession 2013Slide 17

Day 3 – May 9 – WBL Problem #1: Graphical Analysis PC141 Intersession 2013Slide 18 An object has a constant, non-zero acceleration. A graph of position vs. time for this object is: A A horizontal line B A nonhorizontal and nonvertical straight line C A vertical line D A curve WBL LP 2.6

Day 3 – May 9 – WBL Problem #2: Deceleration PC141 Intersession 2013Slide 19 Which of the following is true for a deceleration? A The velocity remains constant B The acceleration is negative C The acceleration is in the direction opposite to the velocity D The acceleration is zero WBL LP 2.11

Day 3 – May 9 – WBL Problem #3: Average Speed PC141 Intersession 2013Slide 20 Your drive 4.00 km at 50.0 km/h and then 4.00 km at 100 km/h. Your average speed for the entire 8.00 km trip is… A Less than 75.0 km/h B Equal to 75.0 km/h C Greater than 75.0 km/h

Day 3 – May 9 – WBL Problem #4: Nerve Conduction PC141 Intersession 2013Slide 21 The human body contains different types of nerves, and the speed at which impulses travel along these nerves strongly depends on the particular type. The sensation of touch relies on Aα receptors, which conduct impulses at roughly 100 m/s. The sensation of pain relies on C receptors, which conduct impulses at about 0.6 m/s. Assume that your left big toe is 160 cm from your brain. What is the time lag between realizing that you’ve stubbed your toe and feeling the resulting pain? Solution: In class

Day 3 – May 9 – WBL Problem #5: Train Trip PC141 Intersession 2013Slide 22 WBL EX 2.7 A train makes a round trip on a straight, level track. The first half of the trip is 300 km, and is traveled at a speed of 75 km/h. After a 0.50-hour layover, the train returns to its original location at a speed of 85 km/h. What is the train’s (a) average speed, and (b) average velocity? Solution: In class