Mechanics This topic will explain the movements of objects. That part of physics, which studies the description of motion, is called kinematics. Will explore how movement can be described and recorded. This will involve defining the so-called kinematic quantities e.g. displacement, velocity and acceleration and how these quantities are calculated in different situations. We will utilize the next few lectures just trying to build up our expertise on this. Once we complete our study of motion we will move on to a study of dynamics, which will involve explaining why movement happens. We will also see how the effect of gravity on the movement of an object leads to the consideration of the energy a body may posses or transfer. Motion in one dimension We will now start the study of motion. In this lecture we will discuss motion in one dimension. We introduce definitions for four fundamental kinematic quantities in terms of which we will describe the motion of objects. Latter we will derive equations of motion for bodies moving in one dimension with constant acceleration. We will then apply these equations to the situation of a body moving under the influence of gravity alone. Kinematic quantities and definitions The four fundamental kinematic quantities we are interested in are: displacement, velocity, acceleration and time. The table shows the said quantities and the associated symbols

Symbol Quantity Magnitude Components

Displacement Distance, s = ! , ! or (,)

Velocity Speed, = (! , !)

Acceleration Acceleration, a = (a!, a!)

t Time

Definition: Displacement is change in position, = ! ! where ! is the final position and ! is the initial position. The arrow indicates that displacement is a vector quantity: it has direction and magnitude. In 1 dimension, there are only two possible directions, which can be specified with either a plus or a minus sign.

Definition: Average Velocity is displacement over total time elapsed. Mathematically: !!

!!!!!!!!!!

= !"#$%&'()(*+ !"#$ !""#$%!& Units: / (1)

Note: The overbar is frequently used to denote an average quantity is always > 0 so the sign of depends only on the sign of .

Graphical interpretation of velocity: Consider 1-d motion from point P (with coordinates si , ti ) to point Q (at sf , tf ). We can plot the trajectory on a graph (see Figure 2.1). Figure 2.1: Graphical interpretation of velocity

Then from Eq, (1) is just the slope of the line joining P and Q. Definition: Instantaneous velocity calculated by taking shorter and shorter time intervals, i.e. taking 0. It is defined mathematically:

lim!!

!! (2)

Ex 2.1: Table 2.1 gives data on the position of a runner on a track at various times. Table 2.1: Position and time for a runner.

t(s) x(m)

1.00 1.00

1.01 1.02

1.10 1.21

1.20 1.44

1.50 2.25

2.00 4.00

3.00 9.00

Find the runner's instantaneous velocity at t = 1.00 s. As a first estimate, find the average velocity for the total observed part of the run. We have,

= =

!.!! !!!.!! !!.!! !!!.!! !

= 4 (3) From the definition of instantaneous velocity Eq. (2), we can get a better approximation by taking a shorter time interval. The best approximation we can get from this data gives,

= =

!.!" !!!.!! !!.!" !!!.!! !

= 2 (4) We can interpret the instantaneous velocity graphically as follows. Recall that the average velocity is the slope of the line joining P and Q (from Figure 2.1). To get the instantaneous velocity we need to take 0, or . When , the line joining P and Q approaches the tangent to the curve at P (or Q). Thus the slope of the tangent at P is the instantaneous velocity at P. Note that if the trajectory were a straight line, we would get v = , the same for all t. Note:

Instantaneous velocity gives more information than average velocity. The magnitude of the velocity (either average or instantaneous) is referred

to as the speed. Definition: Average acceleration is the change in velocity over the change in time: a !!!

!!!!!!!!!!

Units: / ! (5)

Definition: Instantaneous acceleration is calculated by taking shorter and shorter time intervals, i.e. taking 0:

a lim!!

!! (6)

Note: Acceleration is the rate of change of velocity. When velocity and acceleration are in the same direction, speed increases

with time. When velocity and acceleration are in opposite directions, speed decreases with time.

Graphical interpretation of acceleration: On a graph of v versus t, the average acceleration between P and Q is the slope of the line between P and Q, and the instantaneous acceleration at P is the tangent to the curve at P.

From now on velocity and acceleration will refer to the instantaneous quantities.

Derivation of Kinematic Equations of Motion Choose ti 0, si 0, vi v0, and write xf x, vf v and tf t.

a = constant a = a. Then Eq. (5) a = !!!!! or = ! + a (7)

a = constant v changes uniformly = !!(!!!!) From Eq. (1) = s/t .

Combining: s = t =!!(! + )t . Using Eq. (7) we get:

= ! + !!a! (8)

Eq. (7) t = (v - v0)/a. Substitute into Eq. (8) s = (v + v0)(v - v0)/(2a) or,

!! ! ! !! !! !! !! ! ! !!!!!!!!!!!!!!!!!!!!!!!!!! ! ! Note that only two of these equations are independent.

Freely Falling Bodies A freely falling object is an object that moves under the influence of gravity only. Neglecting air resistance, all objects in free fall in the earth's gravitational field have a constant acceleration that is directed towards the earth's center, or perpendicular to the earth's surface, and of magnitude ! g = 9.8 m/s 2. If motion is straight up and down and we choose a coordinate system with the positive y-axis pointing up and perpendicular to the earth's surface, we describe the motion with Eq. (7) Eq. (8) Eq. (9) with a -g, s !y. Equations of Motion for the 1-d vertical motion of an object in free fall:

v = v0 - gt y = v0t - !! gt 2 v 2 = v02 - 2gy

Note: Since the acceleration due to gravity is the same for any object, a heavy object does not fall faster than a light object.

Worked examples In class I promised to give an example of the graphical representation of motion in one dimension.

A displacement vs. time graph is a graph that shows the position of an object along some axis as a function of time. Such a graph is shown in Fig. 2.2 below. The following example relies on your being able to interpret the motion that is described by the graph, and to apply the definitions of average speed and average velocity to numerically describe the motion.

Ex. 2.2 Fig. 2.2 below shows the displacement-time graph for someone walking (or running!) along the x-axis, which has been drawn on the classroom floor. The origin (x = 0) is at an arbitrary position on the floor, and the positive direction has been chosen to point towards the right.

Fig. 2.2

(a) Qualitatively discuss the motion of the person as described by Fig. 2.2.

(b) At what time does the person first come to a stop?

(c) How fast is the person moving at t = 5 s?

(d) What is the persons velocity at t = 12 s?

Solution

(a) Between t = 0 and t = 8 s, the person is moving with a positive velocity (since the slope is positive, or the line slants upwards), meaning that the person is moving towards the right. At t = 8 s, when she is 4.0 m to the right of the origin (since x is positive), the person turns around and starts moving towards the left (that is, in the negative direction, since the slope is negative). At t = 18 s, the person stops moving; she is at the position x = 1 m. She stands there for 4 s, after which she moves rather quickly in the positive direction (larger, positive slope), stopping for just an instant and turning around at t = 30 s, and moving in the negative direction until she stops at t = 36 s and remains at x = 5.0 m until the end of our data at t = 40 s.

(b) a person is stopped when the velocity is zero. From our discussion above, this means that the slope of the graph would be zero. So we can certainly see that the person is stopped during the time intervals from t = 18 s until t = 22 s, and from t = 36 s until t = 40 s. But this involves only the average velocity during a given time interval. It is also possible for the person to stop for only an instant. For example, if you bounce a ball off of a wall, the ball is moving forward, hits the wall and slows down and stops - just for and instant - and then flies away backwards. Whenever an object moving along a straight line (such as the person walking along the x-axis) changes direction, it must stop for at least an instant before it can turn around. Note that the person is moving in the positive direction toward the right (positive slope) from t = 0 s until t = 8 s; then the person is moving toward the left (negative slope) from t = 8 s until t = 18 s. This means that the person had to turn around at t = 8 s. Since she has to stop in order to reverse direction, she stops first at the time t = 8 s.

(c) This is easy in light of our discussion for part (a) above. Note that the slope of the graph between t = 0 and t = 8 s is constant (that is, the graph is a straight line during this interval). Thus, the average speed and average velocity must also be a constant during this interval. If the speed is constant, then the speed at any instant during this interval must equal that constant average speed. The speed at some instant is called (of course!) the instantaneous speed. Am sure you can find the average speed during the interval from 0 to 8 s

(d) Use the formula for calculating instantaneous velocity