Kinematics lesson3 sb

VELOCITY-TIME GRAPHS: UNIFORM AND NON-UNIFORM MOTION KINEMATICS: Lesson 3

The shape of the velocity-time graph reveals whether the object is at rest, moving at constant speed, speeding up, or slowing down.

DATA TABLE

Suppose an airplane has a cruising altitude of10 600 m and travels at a constant velocity of900 km/h [E] for 5.0 h.

TABLE & GRAPH

1. SLOPE OF A VELOCITY-TIME GRAPH

Just as the slope of a position-time graph reveals the rate at which position changes (velocity), the the slope of a velocity-time graph reveals the rate at slope of a velocity-time graph reveals the rate at which velocity changes (acceleration).which velocity changes (acceleration).

SLOPES OF VELOCITY-TIME GRAPHS

The slope of a velocity-time graph represents the acceleration of the object.

Non-uniform Motion – riding a quad

Velocity is not constant.

a. Definition of ACCELERATION An object whose velocity changes (in this case,

increases) over a time interval is undergoing accelerationacceleration, represented by the variable aa.

Acceleration is a vector quantity. vector quantity.

It is also called non-uniform motion because the object’s speed or direction is changing.

When velocity decreases it is undergoing negative acceleration, also represented by the variable a.a.

Let’s determine the slope of the plane’s velocity time graph. Designate east as the positive direction:

Notice the units of the slope of the velocity-time graph: km/h2.

These units are units of acceleration.

Because the plane is moving at a constant velocity, its acceleration is zero.

In general, you can recognize acceleration values by their units.

In physics, the standard units for acceleration are metres per second per second, which is generally abbreviated to m/s2. (read metres per second squared).

b. NEGATIVE ACCELERATION What is the meaning of negative acceleration?

When interpreting the sign of acceleration, you need to compare it to the sign of velocity.

When the directions (signs) of velocity and acceleration are the same (positive or negative), the object is speeding up.

When velocity and acceleration have opposite directions (signs), the object slows down.

2. AREA UNDER A VELOCITY-TIME GRAPH

The area under a velocity-time graph represents the objects displacement.

Unit analysis indicates that the area under a velocity-time graph equals displacement.

Multiplying l x w of each rectangle gives us

For eastward displacement, the area is above the time axis, so it ispositive. For westward displacement, the area is below the time axis, so it is negative.

EXAMPLE 1.6

To calculate displacement we must find the sum of the three areas (A, B and C) under the velocity-time graph.

Designate east as the positive direction (above the horizontal axis) .

Convert minutes to seconds. Let’s do this together!

CALCULATIONS FOR PART A:

b. To find acceleration, find the slope of each section of the graph.

Region C is a horizontal line so its slope is also zero.

To calculate average velocity, determine the area under the velocity-time graph and divide it by the total time.

To find the average velocity given different displacements over different time intervals, add the total displacement and divide by the

total time.

3. AVERAGE VELOCITY ON A VELOCITY – TIME GRAPH

EXAMPLE 1.7

EXAMPLE 1.8

From the graph, determine:

(a) whether acceleration is positive, negative, or zerofor each section

(b) when the bird changes direction.

A bird starts flying south. Its motion is described in the velocity-timegraph in Figure 1.46.

Acceleration is the slope of each section of the graph.

A: NegativeFinal velocity is more negative than the initial velocity, as the bird is speeding up in the south direction. So the slope of this line is negative. The bird’s acceleration is negative.

B: ZeroAcceleration is zero because the slope is zero (the graph is a horizontal line.)

C: NegativeAcceleration is negative because the slope of the line is negative (as in section A).

D: ZeroAcceleration is zero because the slope of the line is zero (as in section B).

E: PositiveFinal velocity is positive because the bird is now flying north. So the slope of this line is positive. The bird’s acceleration is positive.

F: ZeroAcceleration is zero because the slope of the line is zero.

4. INSTANTANEOUS VELOCITY Instantaneous velocity is the velocity of

an object taken at a specific time, an instant.

Earlier in this section we learned that to determine the velocity of an object from a position-time graph you calculate the slope of the graph.

How can you obtain the slope of a curve?

To determine the velocity (and therefore slope) of an object at any instant, physicists use tangents.

A tangent is a straight line that touches a curve at only one point.

INSTANTANEOUS VELOCITY

Each tangent on a curve has a unique slope, which represents the velocity at that instant.

In order for the object to be at that position, at that time, it must have an instantaneous instantaneous velocity equal to the slope of the tangent velocity equal to the slope of the tangent at that point.

You can now create a new table using the slopes of the position-time graphs.

This velocity-time graph represents an object undergoing uniformly accelerated motion.

SLOPE OF A VELOCITY-TIME GRAPH

An acceleration-time graph for an object undergoing uniformly accelerated motion is straight line with zero slope.

SLOPE OF A VELOCITY-TIME GRAPH

EXAMPLE 1.5

This ATV is undergoing non-uniform motion. It is accelerating, in this case, slowing down.

What would the position-time graph look like? What would the velocity-time graph look like? What would the acceleration-time graph look

like?

EXAMPLE 1.5

How do we determine the acceleration?

The acceleration of the ATV is 2.8 m/s2. Because the forward direction was designated as positive, the negative sign means that the direction of acceleration is backward.

SUMMARY Velocity-time graphs are very useful. They

provide the following information: The velocity-time graph gives you instantaneous

velocity values at given times. Calculating the slope of a velocity-time graph gives

you an object’s acceleration. The area under a velocity-time graph gives you the

object’s displacement. You can determine the average velocity of an object

over a time interval from a velocity-time graph.

Homework from Textbook: Check and Reflect

Page 30: Q # 1 – 4 Practice Problems