Reading Guide Chapter 2

Kinematics … the description of motion

Dynamics … the study of causes of motion

2.1 Reference Frames and Displacement

In order to describe the motion of an object we need to make measurements of the position as a function of time. Position measurements are made in a reference frame. We will be using the *Cartesian coordinate system (three axes at right angles to each other, X Y Z). Usually we will restrict our discussions to motion in at most 2 dimensions. Non-Cartesian coordinate systems include: a) Latitude, Longitude, Altitude; b) Spherical Coordinates, etc.

You will often have some freedom in where to place the origin of the coordinate system. Does the placement of the origin affect the results of motion calculations?

*Displacement is the change in position of an object. When working in one dimension the direction of change is indicated by a + or - in front of the size of the displacement. D X = X_f - X_i

A certain person walks 80 cm north, 60 cm south and 30 cm north. *What is the displacement?

Quantities that have both a size (magnitude) and direction are called vectors. Displacement is a vector.

2.2 Average Velocity

*average speed = (total distance traveled) / (time required)

How does distance differ from displacement?

How far did you drive to come to Midland this semester?

How much time was required for the trip?

*Calculate your average speed ____________

*average velocity v = displacement / time required = D X / D t This has a direction.

Will the average velocity value usually be larger, the same size or smaller than the average speed?

Why?

Suppose that two objects (A and B) have the following positions as a function of time.

time A B Construct a rough graph

0 sec 0 m 0 m of position vs. time

1 1 2 for both objects.

2 2 4 Put time on the

3 3 6 horizontal axis.

4 4 8

Use the table of numbers to calculate the

average speed and average velocity for object B.

*To calculate the slope of a straight line: 1. select two points on the line that are far apart, 2. write down the coordinates for the two points, 3. divide the change in the vertical coordinate by the change in the horizontal coordinate.

*Calculate the slope of the line for object B.

2.3 Instantaneous Velocity

*instantaneous velocity, v is the velocity of the object at a particular instant of time.

*The instantaneous velocity can be found by finding the slope of the tangent line on a graph of position vs. time. A tangent line touches the graph at the time of interest with uniform gaps between the graph and the line on both sides of the touch point.

v can be found by calculating D X / D t in the case as D t approaches 0. This requires the use of Calculus and will not be done by the 151 students.

*The size of the instantaneous velocity is called the speed of the object. Speed does not have a direction. Quantities that do not have an associated direction are called scalars. Speed is a scalar.

Suppose that a plane flies north at 150 km/hr. How far does the plane travel in 3 hours?

Under what conditions is the formula distance = rate x time useful?

Make a velocity vs time graph for the plane.

How can the distance traveled be determined from

this graph?

How can the distance traveled be determined

if the velocity vs time graph has this shape?

What is the meaning of a velocity of 5 meters/sec towards the north?

2.4 Acceleration *The measure of the rate of change of velocity is called acceleration. The average acceleration is found by dividing the finite quantities D X and D t.

a_avg = D v/ D t

*The instantaneous acceleration is the acceleration value at an instant in time. The value of the instantaneous acceleration is found by calculating the slope at a particular time on the velocity graph. How would you compare the mathematical connection between velocity and displacement to the connection between acceleration and velocity?

* TRUE or FALSE If the velocity is zero then the acceleration is zero.

* TRUE or FALSE If the velocity is positive then the acceleration is positive.

* TRUE or FALSE If the acceleration is zero then the velocity is zero.

What is the meaning of an acceleration of - 3 m/s² ?

2.5 Motion at Constant Acceleration

In some situations (e.g. vertical motion near the surface of the earth) the acceleration value is constant. The motion is called uniformly accelerated motion. We can easily generate 4 equations which apply to this type of motion.

The acceleration equation similar to d=rt is V = at. To make it more general we can allow the object to have some non-zero initial velocity, v_o , and write:

a = D v / D t or a = (v - v_o ) / t Multiply both sides by t and add v_o to both sides.

For the case of constant acceleration an object has an average velocity found by adding the start and end velocities and dividing by 2. ½ (v + v_o) If the acceleration is not constant would this be the correct calculation for the average velocity?

We know that d = rt applies when r is a constant. *Is rate a constant when the acceleration value is not zero?

Is the average velocity a constant during a certain time interval?

Since the average velocity does represent the motion we can say that

d = rt or

X - X_o = ½ (v + v_o) t or

In some situations you may not know the final velocity, v. In the space provided below, substitute out v in equation 2 by using equation 1. Simplify.

In some situations you may not know the time value. Substitute out t in equation 2 by using equation 1. Simplify.

2.6 Solving Problems

Read through the problem solving guide at least twice. Work through the examples in the text and let me know if you have any questions on the solutions shown in the text.

2.7 Falling Objects

Free fall is vertical motion with only the force of gravity acting on the object. (i.e. ignore air resistance and don’t allow any propulsion)

*g is the symbol for the acceleration due to gravity Gravity is a force, not an acceleration. Gravity produces an acceleration. Near the earth’s surface the value of g is about 9.80 m/s².

Suppose that an object is dropped from rest and is in free fall from a height of 400 feet. Calculate the position of the object for every second before it hits the ground. Equation 2-10b can be used by changing x to y. Use the numbers you create to make a graph of y vs t² . Is this graph a straight line? Does your graph support the text’s statement that x is proportional to t² ?

TRUE OR FALSE The four kinematic equations can only be used in situations in which the acceleration value is constant during the time interval.

2.8 Graphical Analysis of Linear Motion

The slope of a portion of a graph can be found by drawing a straight line tangent to the graph. A tangent line touches the graph at the time of interest and has equal spacing between the tangent line and the graph on both sides of the touch point. See page 39.

The slope of the tangent line on a graph of X vs. time gives information about the instantaneous velocity at that time. The slope of the tangent line on a graph of v vs. time gives information about the instantaneous acceleration at that time.

The area under the graph of velocity vs. time gives information on the change in position of the object. The area under the graph of acceleration vs. time gives information on the change in velocity of the object.

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