Chapter 2 Describing Motion: Kinematics in One...

Chapter 2

Describing Motion: Kinematics in One Dimension

• Introduction

• Reference Frames and Displacement

• Average Velocity

• Instantaneous Velocity

• Acceleration

• Motion at Constant Acceleration

• Falling Objects

(a) Translation(b) Rotation

Motion is part of our dai ly life:

§ Get up in the morning§Walk to the UofR§ Go for a lunch, etc.

If you are at res t and decide to go somewhere, you have to break your inertia and move; changing direction to avoid a crazy teen driving a car also implies motion; etc.

Mechanics is the field in physics dedicated to the study of motion. It is divided into two parts :

Kinematics : Describes how objects move (where an object is at certain given time) Dynamics : Describes the forces responsible for the motion of objects (how they are set, and kept, in motion)

There are two types of motion: Translation and Rotation.

We will start with one-dimensional translational motion: objects that move along a straight-line path.

IntroductionIntroduction

Suppose I say:

“A person walks toward the front of a trai n at 5 km/h. The train is moving at 80 km/h”.

First question we should ask ourselves: The train is at 80 Km/h wi th respect to what?à Let’s assume it is with respect to the ground

Now, what is the speed of this person with respect to me if:

1) I am at rest on the train2) I am at rest on the ground at a railway station

This example shows the importanceof identifying a frame of reference (or reference frame) with respect to ityou can define your motion.

In our example, if you consider the person moving on the train:

- It’s speed is 5 Km/h if the reference frame is defined as the train; - Its speed is 85 Km/h if the reference frame is defined as the ground.

Reference Frames and DisplacementsReference Frames and Displacements

à 85 Km/hà 5 Km/h

Ground

In the previous example, if I ask you the ques tion: “Is the train going from Calgary to Vancouver at 80 Km/h with respect to the ground? ”, will you be able to answer?

à No, because you do not know the direction it is moving (Calgary to Vancouver or vice-versa?)

Despite the fact that we have identified the ground as the reference frame with respect to it the speed of the train is measured, we have not ful ly characterized the motion of the train. We also need to specify its direction.

The direction can be specified by defining a coordinate system which represents the reference frame used to study the motionof an object.

In our previous example, we can choose a point along the rai lroad line as the origin, or zero “0”, of our coordinate system. We then draw two perpendi cular axes, x and y, crossing at the origin of the system.

But, how do we use such a system to specify direction?


Definitions:

• Objects located to the right of the origin on the x axis have positive x (+x) coordinates.

• Objects located to the left of the origin on the x axis have negative x (-x) coordinates.

• Objects located above the origin on the y axishave positive y (+y) coordinates.

• Objects located below the origin on the y axis have negative y (-y) coordinates.

A point P in this coordinate system is fully identified by a pair of coordinates (x,y)

For one-dimensional motion, we usually choose the x axis as the line of motion of an object. If the motion is vertical, we usually choose the y axis.


P(x,y)

x

y

Some notes:

The definitions of the signs on the previous slideare arbitrary. You can choose your own signconvention as long as you use it consistently while solving a problem.

For example:

• you could have chosen positive x positions to be located on the left side of the origin;

• or have chosen positive y positions to be located below the origin.

We will see some examples later on.

Reference Frames and DisplacementsReference Frames and Displacements+ y

- x

- y

+ x

- y

- x

+ y

+ x

Back to our example, let’s now locate the initial and final positions of the train in the coordinate system we have just defined. Note that these two positions correspond to measurements taken at two different times that we will call t1 and t2. Let’s then call them positions x1 for t1 and x2 for t2.

Note: Positions x1 and x2 are measured relative to a point on the train (say, front of the train) in our coordinate system.


x1 x2

x1 = Calgary

x2 = Vancouver

Now we know that the train is going from x1 to x2, so we have a direction.

Say x1 represents Calgary and x2represents Vancouver. You are now able to tell me which way the train is travelling:

à Calgary to Vancouver.

In the previous slide, the train has been displaced from its initial position x1 to its final position x2. We define this displacement as:

Note: Here the symbol Δ denotes “change”.


x1 x2

Note that Δx is not only a number(scalar) but also carries informationabout direction. We can compile both information by defining a vector:

Where the arrow on the top of Δxdefines a vector.

The magnitude of (2.2) is given by (2.1).

(2.1)

(2.2)


Examples:

1) Suppose an object moves from position x1measured at an initial time t1 to position x2 at a final time t2 as shown is figure (a). The blue arrow represents the displacement x2 – x1, or the vector

Suppose x1 = 10 m and x2 = 30 m. Then, using (2.1):

Δx is positive denoting a vector in the positivex direction (right direction).

2) If we invert the initial and final positions such that x1 = 30 m and x2 = 10 m as shown in figure (b), then, again using (2.1):

Now Δx is negative denoting a vector in the negative x direction (left direction).

(a)

(b)


Note:

You shall not confuse displacement with distance travelled.

à Displacement gives how far distant an object has been displaced from its starting point, along with the direction of such displacement. It is a vector quantity.

In the figure below, the starting position is at x1 = 0 m and the final position is at x2 = 40 m. The displacement is:

regardless whether the object had to travel to x3 = 70 m before reaching x2 = 40 m.

à Distance gives the total length an object travels from its starting position, regardless the direction of its motion. It is a scalar quantity.

The total distance is given by: x1 x2 x3

Average VelocityAverage Velocity

We usually make no distinction between the terms speed and velocity. However, in physics they refer to di fferent quantities and their distinction is very important.

Definitions:

Average Speed: Refers to how far an obj ect travel in a given time interval and is related to total distance.

Speed is a scalar (do not depend on the direction of motion).

Example: In our previous example, an object moves from x1 = 0 m to x2 = 40 m passing by x3 = 70 mbefore reaching x2. Assuming the time elapsed to go from x1 to x2 is given by Δt = t2 – t1 = 5 s, the averagespeed is:

(2.3)

x1 x2 x3


Average Velocity: Gives not only how fast an object travels but also its direction of motion. Contrary to average speed, average velocity is related to displacementinstead of total distance traveled. Its magnitude is given by:

or

Note: The bar over v in (2.4) denotes “average”.

Velocity is a vector. The sign of Δx in (2.4) defines the direction of motion. If Δx < 0, the average velocity is negative and therefore the object is moving toward negative xcoordinates, and vice-versa.

(2.4)


Examples:

1) Still using the example pictured in the figure (a),an object is displaced from x1 to x2. Usingequation (2.4), the average velocity is:

The calculated average velocity is positive, therefore the object is moving towardpositive x coordinates.

Note that, for the given example, the calculated magnitude of the average speed (20 m/s) is bigger than that for the average vel ocity (8 m/s).

x1 x2 x3

(a)


Examples:

2) The position of a runner as a function of time is plotted as moving along the x axis of a coordinate system. During a 3.00 s time interval, the runner ’s position changes from x1 = 50 m to x2 = 30.5 m, as shown in figure (b). W hat was the runner ’s average speed? And its average velocity?

Average velocity:

Note: The absolute magnitude of both the average speed and average vel ocity are the same. This is always true when the moti on is all in one direction.

(b)

Instantaneous VelocityInstantaneous Velocity

So far we have used the term “average” for velocity and speed. This is due to the fact that the velocity can change along the path of moti on of an object, though we can always assign an average velocity (or speed) to it knowing the time elapsed for the object to go from a initial to a final position.

For example:

à Figure (a) depicts a car moving at constant velocity. In this particular case, the average velocity is similar to the velocity of the car at any instant of time.

à On the other, figure (b) represents a car moving at different velocities at different instant of time. In this case, the average velocity is not necessarily equal to the car velocity at a particular instant of time.

a) constant velocity.

(b) varying velocity.

Instantaneous VelocityInstantaneous Velocity

We can use eq. (2.4) to defi ne the velocity at any instant of time, or as we call: instantaneous velocity. For this purpose, we can make the time interval very short. In fact, we can make it so small that it will be close to zero. We say that Δt is tending to zero, or that i t is an infinitesimally short interval of time.

Note that I have dropped the bar on the top of v.So, the magni tude of the instantaneous velocityis represented by v in (2.5).

From now on, every time I use the term velocityit will refer to the instantaneous velocity v insteadof the average velocity, unless otherwise stated.

a) constant velocity.

(b) varying velocity.

(2.5)

Δx

AccelerationAcceleration

à Acceleration gives how fast the velocity of an object is changing.

The average acceleration is defined as the change in velocity divided by the time taken to make this change:

As velocity, acceleration is also a vector. It can have either positive or negative sign:

Ø If v2 > v1 , Δv > 0 implies that the acceleration vector points towards positive x axis

Ø If v2 < v1 , Δv < 0 implies that the acceleration vector points toward negati ve x axis

Note that acceleration has the units (SI) of m/s2.

(2.6)


An object can go faster (accelerate) or slow down (decelerate) depending on the relative direction between the velocity and acceleration vectors.The two examples below illustrated these concepts:

1) An car moves to the right of a coordinate system as shown in figure (a) (toward positive x axis). The car slows down from an initial velocity v1 = 15.0 m/s to a final velocity of v2 = 5.0 m/s in a time interval of 5.0 s. What is the car’s average acceleration?

Using eq. 2.6,

Note that the acceleration is negative and pointsin the opposite direction of the velocity.

Note also that if we invert the initial and final velocities such now v1 = 5.0 m/s and v2 = 5.0 m/s , the resulting average acceleration will have the same magnitude but with a positive sign pointing in the same direction as the velocity vector. Therefore, in this case we have an acceleration.


Example: In this figure, the car is accelerated from a velocity v1 = 0 Km/h to v2 = 75 Km/h at a rate of 15 Km/h per second.


Similarly to instantaneous velocity, we can define instantaneous acceleration as:

.

(2.7)

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