# VECTORS AND BASIC MOTION

Dr. Johnny B. Holmes

## Introduction

In this first part of the course, we consider: 1. what physics is; 2. the concept of vectors, and 3. the basic description of motion. We first of all consider what physics is. A first attempt at a definition might be this: Physics is the science that considers the basic structure of matter, the basic properties of matter, the basic interactions between pieces of matter, and the basic descriptions of motion. Physics attempts to describe as many natural phenomena (happenings) as it can in terms of as few basic principles (laws) as it can.

Note: Natural phenomena are usually very complex things and, to start with, we will be making many idealizations and simplifying assumptions. Once the basic principles are known, you can begin to consider removing some of these simplifying assumptions and try to obtain more and more accurate descriptions of real phenomena.

Note: This is a physics course and not an applications course. We will consider some applications as examples, but the applications are not the primary aim of the course. They are only a very nice side benefit. We are primarily seeking the basic structure, properties, interactions, and descriptions. You can deal with more applications in engineering courses or on your own (if you understand the basic principles, you should be able to do this), but time puts a severe limit on our discussion of applications in this class. If you wish to talk about applications outside of class, I will be glad to talk to you!

Physics is a science which means that it involves measurements. We discuss measurements and the associated units and dimensions. This leads to the metric system.

Next we discuss the idea of vectors and coordinate systems in anticipation of the ideas of force and motion. You must learn to think in terms of components of vector quantities if you are to work successfully in physics (and hence, in engineering). Our first lab experiment deals with these ideas.

Finally we consider the basic description of motion: position, velocity, and acceleration. You must be able to distinguish velocity and acceleration - they are not the same! Our second lab experiment deals with this.

## Physics and Measurement

OUTLINE:

1. Introduction
1. definition of physics
2. review of math [algebra, simultaneous equations, basic trig]
3. measurements, dimensions and units
4. the metric system [meters, kilograms, seconds]
5. dimensional analysis
6. order of magnitude estimates

SUPPLEMENTARY HOMEWORK PROBLEMS (S-):

1. Given the following three equations, solve them for x, y, and z:

ax + by - cz = 5 where a,b,c are the last three digits of your phone number (e.g., 321-3448 means a=4, b=4, c=8);

-dx + ey + fz = 8 where d,e,f are the last three digits of your (or your parent's) street address or box number

gx - hy - kz = 0 where g,h,k are the last three digits of your (or your parent's) zip code.

HINT: in the look back stage (step 7), check your answers by substituting in your answers into the equations to show that they indeed work.

## Directed Quantities: Vectors

OUTLINE:

1. Positions
1. lengths
2. angles q (in radians) = s/r [where s is arclength]
3. 2-D coordinate systems
1. rectangular (x,y) [most basic - due to addition properties]
2. polar (r,q ) [most common - magnitude, direction]
3. lattitude and longitude [on curved surface of earth]
4. transformations from one form to the other:
1. polar to rectangular: x = r cos(q ); y = r sin(q )
2. rectangular to polar: r = Ö (x² + y²); q = tan-1(y/x)
2. Vectors
1. idea of a vector: magnitude and direction (position is example)
2. components of a vector (MOST IMPORTANT!)
1. rectangular (x and y components)
2. polar (magnitude and direction)
3. three (or more) dimensions
3. Force
1. the idea of force: a push or pull
2. force is a vector: it has magnitude and direction
3. forces can be added as vectors to get resultant force

SUPPLEMENTARY HOMEWORK PROBLEMS (S-):

2. If the arc length is 3.0 meters and the radius is 12 meters, what is the angle: a) in radians? b) in degrees? c) in revolutions?

3. The moon has a diameter of 3,476 km and is 384,500 km away. What angle does the moon make with a person's eye: a) in radians? b) in degrees? c) in cycles? [Hint: DRAW A GOOD DIAGRAM showing both the given distances and the angle at the eye.]

4. What is the displacement of the point of a wheel initially in contact with the ground when the wheel rolls forward 3/4 of a revolution? (The radius of the wheel is 'R' and the 'X-axis' is the forward direction.) (HINT: break the motion into two part: the translation of the wheel and the rotation of the wheel. Only look at initial and final points, not the actual trajectory.)

5. A car drives five blocks East, turns North for two blocks, then turns back West for 2 blocks. What is the final position of the car relative to the initial position. Express in both rectangular and polar form.

2. a) 0.25 radians, b) 14.32°, c) 0.0398 revolutions.

3. a)9.04 x 10-3 radians; b) 0.52°; c) 1.44 x 10-3 cycles.

4. (5.71*R, 1.00*R) or (5.80*R, 9.9°)

5. (3 blocks East, 2 blocks North) or (3.61 blocks, 33.7° North of East)

## Motion in One Dimension

OUTLINE

1. Introduction: Motion is change in position with time
2. Velocity: change in position with time
1. average velocity: vx-avg = Dx / Dt,
2. use with DISCRETE DATA POINTS & NUMERICAL METHODS (computers!)

3. velocity is a vector: magnitude (speed) and direction, but work in rectangular!
4. instantaneous velocity vx-inst = dx(t)/dt [calculus derivative], use with FUNCTIONS
3. Acceleration: change in velocity with time
1. average acceleration: ax-avg = Dvx / Dt
2. use with DISCRETE DATA POINTS & NUMERICAL METHODS (computers!)

3. acceleration is a vector: magnitude and direction, but work in rectangular!
4. instantaneous acceleration ax-inst = dvx(t)/dt (another derivative) use with FUNCTIONS
4. Going backward: finding vx from ax, x from vx
1. Dvx = ax-avg Dt, or vx-final = vx-init + ax-avg t
2. Dx = vx-avg Dt, or xfinal = xinit + vx-avg t
3. for functions, can use calculus (inverse of derivative is integral)
5. A useful special case: constant acceleration
1. formulas: x(t) = xo + vot + 1/2at² ; v(t) = vo + at
2. freely falling bodies: ay = -9.8 m/s² (+ means up, - means down; freely means with negligible air resistance)
6. Motion graphs
1. getting velocity from position [slope of x(t) Û value of v(t)]
2. getting acceleration from velocity [slope of v(t) Û value of a(t)]
3. getting position from velocity [value of v(t) Û slope of x(t)]
4. getting velocity from acceleration [value of a(t) Û slope of v(t)]

SUPPLEMENTARY HOMEWORK PROBLEMS (S-):

6. Below is a graph of x(t). On the graphs below it, sketch v(t) and a(t).

7. Below is a graph of ax(t). On the graphs below it, sketch x(t) and vx(t) assuming that xo > 0 and vxo < 0.

8. Below are the numerical values of velocity at specific times as well as the functional expression for velocity. a) Using the numerical method calculate both the average acceleration during the time between t=8 and t=9 seconds, and calculate the position at t=9 seconds. You should assume that x=-20 m when t=0 [i.e., xo=-20m]. b) To check yourself, use the functional form for a(t) and x(t) given below (this was derived using calculus assuming xo=0). Specifically, evaluate the acceleration at t=8.5 sec and compare to the average acceleration between t=8 and t=9 sec., and evaluate the position at t=9 sec and compare to what you got using the numerical procedure. HINT: remember for the numerical method you use:

vavg = Dx/Dt, and aavg = Dv/Dt where Dx = xi+1 - xi , Dt = ti+1 - ti .

NUMERICAL DATA:

 v (0 sec) 50.0 m/s * v (6 sec) 24.8 m/s v (1 sec) 49.3 m/s * v (7 sec) 15.7 m/s v (2 sec) 47.2 m/s * v (8 sec) 5.2 m/s v (3 sec) 43.7 m/s * v (9 sec) -6.7 m/s v (4 sec) 38.8 m/s * v (10 sec) -20.0 m/s v (5 sec) 32.5 m/s *

FUNCTIONAL FORMS:

v(t) = 50 m/s - (0.700 m/s3)t²

a(t) = (-1.400 m/s3)t

x(t) = xo + (50.0 m/s)t - (0.233 m/s3)t3

9. For the situation in problem 8 above, graph v(t) versus t. From this graph, be able to qualitatively graph x(t) versus t, and a(t) versus t.

10. A car accelerates (assume uniformly) from rest with an acceleration of 1.8 m/s². a) How long a time will it take for the car to reach a speed of 25 m/s ? b) How far will the car have gone in this time? c) How fast will the car be going after 10 seconds? d) How far will the car have gone after 10 seconds?

11. A ball is thrown upwards from the top of a building 10 meters high with an initial speed of 30 m/s. a) How long will it take the ball to reach it's highest point? b) How high will this highest point be? c) How long will it take the ball to hit the ground (at the bottom of the building)? d) How fast will the ball be going when it hits the ground?

12. An airplane on an aircraft carrier takes off from rest and needs to reach a speed of 80 m/s (80 m/s = 180 mph, but the wind and the speed of the carrier make this an effective speed of 225 mph). The carrier has a deck 110 meters long for the take-off. Assume the acceleration of the plane during take-off is constant. a) What is the acceleration necessary for the plane during taking-off? b) What is this acceleration in 'gees' ? c) How long in duration is the take-off?