1. Ch 2 Describing Motion: Kinematics in One Dimension

2. Chapter 2
The study of motion of objects, and the related concepts of force and energy, form the field called mechanics
Mechanics is customarily divided into two parts:
Kinematics which is the description of how objects move
Dynamics deals with force and why objects move as they do
Translational motion deals with objects that move without rotating
We will often use the concept, or model, of an idealized particle which is considered to be a mathematical point and have no spatial extent (no size)

3. Ch 2 cont A particle can undergo only translational motion
The particle model is useful in many real situations where we are interested only in translational motion and the object's size is not so significant

4. 2.1 Reference Frames and Displacement
Any measurement of position, distance, or speed must be made with respect to a reference frame, or a frame of reference
i.e. If a train is moving at 80 km/h and a passenger is walking towards the front at 5 km/h. This 5km/h is the person's speed with respect to the train as a frame of reference.
With respect to the ground the person is moving at 80 km/h + 5km/h = 85 km/h
It is always important to specify the frame of reference when stating a speed

5. 2.1 cont
When specifying the motion of an object, it is important to specify not only the speed but also the direction of motion
We often draw a set of coordinate axes, to represent a frame of reference
For one-dimensional motion, we often choose the x axis as the line along which the motion takes place
Then the position of an object at any moment is given by the x coordinate
If the motion is vertical, as for a dropped object, we usually use the y axis

6. 2.1 cont Displacement is the change in position of the object
How far the object is from the starting point
Distance vs Displacement
If someone walks 70m to the east and turns around and walks a total of 30m, the distance traveled is 100m and the displacement is only 40m
Displacement is a quantity that has both magnitude and direction
Such quantities are called vectors, and are represented by arrows in diagrams
Vectors which point in one direction will have a positive sign, whereas vectors that point in the opposite direction will have a negative sign, along with their magnitude

7. 2.1 cont
∆X = X2-X1
Where delta x means change in X or change in position, which is the displacement
Note: the "change in" any quantity means the final value of that quantity minus the initial value

8. 2.2 Average Velocity
Speed refers to how far an object travels in a given time interval, regardless of direction
The average speed of an object is defined as the total distance traveled along its path divided by the time it takes to travel this distance
Speed vs Velocity
Speed is a positive # with units
Velocity is used to signify both magnitude (numerical value) of how fast an object is moving and also the direction in which it is moving
Thus, velocity is a vector

9. 2.2 cont
There is a second difference between speed and velocity: namely, the average velocity is defined in terms of displacement, rather than total distance traveled
Average velocity = displacement/time elapsed = (final position – initial position)/time elapsed
Average speed and average velocity have the same magnitude when the motion is in one direction
In other cases they may differ
This difference between the speed and the magnitude of the velocity can occur when we calculate average values

10. 2.2 cont
The average velocity, defined as the displacement divided by the elapsed time can be written as:
V= (x2 – x1)/(t2 – t1) = ∆X/ ∆t
V with the line above it stands for average velocity, the line always symbolizes average
The elapsed time, or time interval, t2 – t1, is the time that has passed during our chosen period of observation
The direction of the average velocity is always the same as the direction of the displacement

11. 2.3 Instantaneous Velocity
Instantaneous velocity is the velocity at any instant of time. More precisely, it is defined as the average velocity during an infinitesimally short time interval
We define instantaneous velocity as the average velocity as we let ∆t become extremely small, approaching zero
We can write the definition as
The notation lim delta t --> 0 means the ratio of delta X/delta t is to be evaluated in the limit of delta t approaching zero
For instantaneous velocity we use the symbol v, whereas the average velocity we use v, with the bar

12. 2.3 cont
The instantaneous speed always equals the magnitude of the instantaneous velocity
Why?
Bc the distance and the magnitude of the displacement become the same earn they become infinitesimally small
If an object moves at a uniform (constant) velocity during a particular time interval, then it's instantaneous velocity at any instant is the same as its average velocity
But in reality this is not the case

13. 2.4 Acceleration
An object whose velocity is changing is said to be accelerating
Acceleration specifies how rapidly the velocity of an object is changing
Average acceleration is defined as the change in velocity divided by the time taken to make this change
average acceleration = change of velocity/time elapsed
a = (v2 – v1)/(t2 – t1) = ∆v/∆t
Acceleration is also a vector, but for one-dimensional motion, we need only use a + or – sign to indicate direction relative to a chosen coordinate system

14. 2.4 cont
The instantaneous acceleration a can be defined in analogy to instantaneous velocity for any specific instant
a = lim∆t→0 ∆v/∆t
We almost always write the units for acceleration as m/s2
This is possible bc m/s/s = m/(s*s) = m/s2
Note: Acceleration tells use how quickly the velocity changes, whereas velocity tells us how quickly the position changes

15. 2.4 cont
Deceleration does not mean that the acceleration is necessarily negative
For an object moving to the right along the positive x axis and slowing down, the acceleration is negative
But the same car moving to the left (dec x) and slowing down, has positive acceleration that points to the right
We have a deceleration whenever the magnitude of the velocity is dec, and then the velocity and acceleration point in opposite directions

16. 2.5 Motion at Constant Acceleration
Many practical situations occur in which the acceleration is constant or nearly constant
When the magnitude of the acceleration is constant and the motion is in a straight line. The instantaneous and average accelerations are equal
v = (x-x0)/(t-t0) = (x-x0)/t , where t is the elapsed time
a = (v-v0)/t

17. 2.5 cont
A common problem is to determine the velocity if an object after any elapsed time, t, when we are given the object's constant acceleration
v = v0 + at constant acceleration
How to calculate the position of an object after a time t when it is undergoing constant acceleration
x = x0 + vt
Bc velocity inc at a uniform rate, the average velocity v, will be midway btwn the initial and final velocities
v = (v0 + v)/2 constant acceleration
Careful this is not necessarily valid if the acc is not constant

18. 2.5 cont We combine the last three equations to find
x = x0 + vt = x0 + [(v0+v)/2]*t
= x0 + [(v0 + v0 + at)/2) * t
OR x = x0 +v0t +1/2 * at2 constant acceleration
Where t is unknown
v2 = v02 + 2a(x-x0) constant acceleration

19. Useful Eq. When a is Constant
v = v0 + at
x = x0+v0t + (1/2)at2
v2 = v02 + 2a(x-x0)
v = (v + v0)/2
These eq are not valid unless a is constant
In many cases we can set x0 = 0
NOTE: x represents position, not distance, that x – x0 is the displacement

20. 2.6 Solving Problems