Science/Motion

= Motion in One Dimension = '''

In this unit, we will attempt to understand and describe motion and explain the motion of objects. Kinematics is the science of describing the motion of objects using words, diagrams, numbers, graphs, and equations. We have already seen that the study of Physics is about building models and a framework to explain the phenomena we observe. If we need to build a model, we need to develop a common vocabulary to describe our observations. Once we have the vocabulary, we can use that to analyze the motion of objects and ultimately explain how they move in the way they do.

In this unit, we will study Motion in One Dimension and the equations of motion that describe the case of motion with uniform acceleration.

= Concept Map = This concept map needs some work. Please refine it and share on the mathssciencestf@googlegroups.com.

= Distance and displacement = In the world around us, we see different objects around us. What would be one of the first observations you would make about an object? The first thing we would notice (other than the shape, size, color, etc.) is whether it is moving or at rest. We often perceive motion (movement) when something changes its position with respect to time. Alternatively, we can infer that something has moved after observing its surroundings. Can you think of examples that describe both these?

When we begin to describe motion, the first thing that comes to mind is that of direction. It is not enough to know that something has moved; it is equally important to know where it has moved. There are many quantities in Physics that get their relevance from the direction in which they are operating.

When a quantity can be described just by its magnitude, it is called a scalar. For instance, mass. It is not very meaningful to say, that body A has mass 20 kg in the eastern direction. On the other hand, I do need to know which is the path to take from Town A to Town B. The description of the path from A to B is meaningful when I know the direction of travel.

* Scalars are quantities which are fully described 	by a magnitude (or numerical value) alone. Mass, volume, area, 	temperature, etc. are scalars.'' * Vectors are quantities which are fully described 	by both a magnitude and a direction. This notion of rest and motion is very relative. In everyday instances, we have several examples of this relative movement with respect to another. (people inside a bus move with respect to those standing on the ground, trees appearing to move backwards when seen from a train moving forward). The next question that comes is what we are measuring when we describe motion. There are several terms that we need to define to describe motion. These are distance, displacement, speed, velocity, acceleration, uniform, non-uniform, etc.
 * Rest : When a body does not change its position 	with respect to surroundings, then it is said to be in a state of 	rest.
 * Motion: When a body changes its position with 	respect to surroundings, then it is said to be in a state of motion.'

'' The physics teacher has worked hard and traveled a distance of 12 m but her effective displacement from the initial position, A, is zero. ''
 * Distance: Distance is the length of the path 	traveled in a certain time. 
 * Displacement: Displacement is the shortest distance between 	the initial and final position.

Since, we are only interested in the initial and final positions for displacement, we are interested in the final position with respect to the initial position – to the left of, to the right of, in front of, etc.



''What is the effective displacement and the distance traveled starting at	t = 0 minutes?''

That brings us to the idea of direction. A measure of displacement makes sense only when we have the direction in which the displacement occurs. Also associated with the idea of displacement is the notion of a reference point, the initial position with respect to which the displacement occurs. Motion can occur in a number of complex ways – straight line, circular, or any combination thereof. We will discuss in this module straight line and circular motion.

= Speed and velocity = Closely associated with the definitions of distance and displacement is that of speed and velocity. Obviously, if an object has covered a certain distance, it has done so over a certain time. This time over which the distance is covered will be different for different objects.

We describe this rate of motion of objects in terms of its speed.

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 * Average speed = Total distance traveled/ Total 	time taken
 * If an object travels a certain distance, s, in 	time t, then the speed, v, is given by v = s/t.

An object travels 16 m in 4 seconds and then another 16 m in 2 seconds. What is the average speed of the object?

Total distance	 = 32 m

Total time		= 6 seconds

Average speed	= 32/ 6 = 5.33 m/s

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We have already seen that direction is needed to describe any motion meaningfully. The change in displacement of an object, in time t, can be expressed as its velocity.


 * Velocity is defined as (final position – 	initial position)/ time taken. Velocity is a vector quantity and 	needs both magnitude and direction. 
 * Velocity describes the speed and the direction in which the motion has occurred.[[Image:Motion%20in%20One%20Dimension_html_m30eb9e7a.png]]

When evaluating the velocity of an object, one must keep track of direction. It would not be enough to say that an object has a velocity of 55 mi/hr. One must include direction information in order to fully describe the velocity of the object. For instance, you must describe an object's velocity as being 55 mi/hr, east. This is one of the essential differences between speed and velocity. Speed is a scalar quantity and does not keep track of direction; velocity is a vector quantity and is direction aware.

The task of describing the direction of the velocity vector is easy. The direction of the velocity vector is simply the same as the direction which an object is moving. It would not matter whether the object is speeding up or slowing down. If an object is moving rightwards, then its velocity is described as being rightwards. If an object is moving downwards, then its velocity is described as being downwards. So an airplane moving towards the west with a speed of 300 mi/hr has a velocity of 300 mi/hr, west.

It is important to understand that an average speed or average velocity does not suggest that the entire distance was traveled at that speed or that the entire displacement occurred at that rate. An average only considers the overall distance (displacement) over a give time interval and calculates the average speed or velocity.

Since a moving object often changes its speed during its motion, it is common to distinguish between the average speed and the instantaneous speed. The distinction is as follows.

If an object is moving in a single direction, the speed could be either constant, changing uniformly (constantly) or changing in a non-uniform way. An object’s velocity can change either in terms of magnitude (the rate at which actual physical units of space are covered) or the direction in which the motion has occurred. If the velocity or the speed is changing at a uniform rate, then the average speed or velocity is given by the arithmetic mean.
 * Instantaneous Speed - the speed at any given 	instant in time.
 * Average Speed - the average of all instantaneous 	speeds; found simply by a distance/time ratio.
 * Average velocity - the total displacement 	divided by the time over which the displacement occurs
 * Instantaneous velocity - the average velocity 	calculated for a very small time interval 

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 * Average velocity 	= (initial velocity + final 	velocity)/ 2
 * Average speed 	= (initial speed + final speed)/ 	2
 * The units of speed and velocity are m/s.

Check your understanding

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 * 1) An object has moved through a 	distance. Can it have zero displacement?
 * 2) A farmer moves along the 	boundary of a square field of side 10m in 40 seconds. What will be 	the magnitude of displacement of the farmer at the end of 2 minutes 	20 seconds?
 * 3) Which of the following is true 	for displacement?
 * 4) It cannot be zero
 * 5) Its magnitude is greater than 		the distance traveled by the object.
 * 6) Usha swims in a 90 m long pool. 	She covers 180 m in one minite by swimming from one end of the pool 	to another and back along the same straight path. Find her average 	speed and velocity.
 * 7) Under what conditions is the 	magnitude of average velocity of an object equal to its average 	speed?
 * 8) What does the odometer of an 	automobile measure?
 * 1) What does the odometer of an 	automobile measure?

= Rate of change of velocity = Let us go back to our Physics teacher walking along the boundary. She may cover 4m East in 2 seconds; 2m South in 2 seconds, 4 m West in 4 seconds and 2 m North in 1 second. Her speed and velocity on each of these sectors is different.

A moving object may not always move at the same rate not does it always have to move at variable rates. An object moving at constant velocity is an example of uniform motion. Velocity varies with time in non-uniform motion. Most objects have a combination of uniform and non-uniform motion. Your drive from school to home is an example of non-uniform motion. Can you think of an object with a uniform motion?

Once we know that an object has a variable velocity, we need to find out how much does the velocity has changed. We do this with by measuring the acceleration, defined as:

Since acceleration is change in velocity in a given time, its units should be m/s per second, m/s 2 .
 * Acceleration = Change in velocity / time taken 
 * If the velocity of an object changes from “u” 	to “v” in time t, then the acceleration is given by a = (v – 	u)/ t.

The acceleration can be uniform or non-uniform. Uniform acceleration means the velocity changes at the same rate. Non-uniform acceleration means velocity does not change at a uniform rate.



What is the direction of the acceleration vector?

If an object is moving in a line and its speed is increasing, the change in velocity is positive. The acceleration in this case is adding to the velocity and is considered to be positive and in the same direction as then velocity vector. What happens when you pedal faster on a bicycle? Which one of these tables represent positive acceleration?

If an object is moving in a line and its speed is decreasing, the change in velocity is negative. The acceleration in this case is reducing the velocity and is considered to be negative and in the opposite direction as then velocity vector. What happens when you apply the brakes on a car?

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Check your understanding

1. In everyday life, you come across a range of motions in which

Can you identify examples?
 * Acceleration is in the direction 	of motion
 * Acceleration is against the 	direction of motion
 * Acceleration is uniform
 * Acceleration is non-uniform

2. Starting from a stationary position, Rahul pedals his bicycle to attain a velocity of 6 m/s in 30 seconds. Then he applies the brakes in such a way that the velocity of the bicycle comes down to 4 m/s in the next 5 seconds. Calculate the acceleration of the bicycle in both the cases.

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'''Acceleration on a freely falling body '''

One example of uniform acceleration is acceleration due to gravity. The force of gravity acts on all objects and they will accelerate downwards if they are allowed to fall. For a freely falling object, this uniform downward acceleration is called acceleration due to gravity and is given by 9.8 m/s2. For a freely falling object, what will happen to the velocity as it falls, ignoring any air resistance?

= Graphs to represent the above quantities = Distance time graphs

So far, we have examined several terms concepts associated with one dimensional motion. It is easier to represent all these relationships in graphs. Motion can be represented as line graphs.



The position of the object is plotted for time from 0 to 5 seconds. The position is at 5 units all through the 5 seconds.

Has the object moved? What is its velocity?

Considerthe object below.



It is covering 0 to 50 m in a time span of 0 – 5 seconds.

The object is staring from 0;jumps to 10 m at t = 1 second and stays there till t = 2 seconds; jumps upto 20 m at t = 2 seconds and so on. Instead of jumping in steps if the object were to move continuously, the position time graph would like the one below.

A straight line means a uniform slope.

Slope = y2 – y1 / x2 = x1     =	change in position/ change in time = velocity

The object is covering equal distances in equal intervals of time; what can you say about the velocity?



In this graph, the object is moving uniformly till t = 3 seconds, when it covers a distance of 2m. After that, it stays at rest.

Can there be a position – time graph parallel to the y-axis? Why or why not? Considerthe object below.





The object here is moving from position 0 m to 50 m in 5 seconds. But it is not covering the same distance in each second.

Slope 		= (y2 – y1 )/ (x2 - x1) = change in position/ change in time = velocity

The rate at which the position is changing is not constant.; this means that the slope at different points is different. What can you say about the velocity?

Can you interpret the following position time graphs?

Velocity - time graphs

In a similar manner, we can also plot velocity time graphs to describe motion of a body.

Examine the graph below. What can you say about the motion? Is the body moving?

Short quiz :

What is the displacement of the body at the end of 3 seconds?


 * 5
 * 15 m
 * Can’t say – not enough information

The graph describes the motion of a body which moves at a constant velocity. At each second, the body covers 5m. Therefore, at the end of 3 seconds, the displacement would be 15 m.

Examine carefully the position – time information below.

Can you draw a velocity-time graph?

What can you say about the velocity of this body?

Is it uniform, increasing or decreasing?

Now, let us consider the position – time information below.





What can you say about the velocity of this object? The velocity is increasing with time. The slope of this graph is calculated as

Slope = (v2 – v1)/ (t2 – t1)

= acceleration

If the slope is the same at all points, we have a case of uniform acceleration. If the slope is uniform and zero, we have a case of uniform, zero, acceleration, which means constant velocity. If the slope is different at different points, we have variable velocity.

What can you say about the acceleration in the two graphs above?

When the acceleration is negative, how will the velocity – time graph look like?

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Check your understanding

1. Consider the graph at the right. The object whose motion is represented by this graph is ... (include all that are true):

From this data, can you calculate the distance traveled by the car?
 * moving 	in the positive direction.
 * moving 	with a constant velocity.
 * moving 	with a negative velocity.
 * slowing 	down.
 * speeding 	up.
 * moving 	with a positive acceleration.
 * moving 	with a constant acceleration.
 * 1) The 	velocity of a moving car is recorded at various instants as follows. 	Can you plot the velocity – time graph?

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'''Distance from a velocity – time graph'''

Consider the following graphs.

The motion of any body with an uniform acceleration can be described starting from the above two equations.

s = (u + v) * t / 2

a	= (v – u)/ t

Starting from these two equations and re-arranging, we get the following equations.

s	= ut + ½ at2

s	= vt – ½ at2

v2	= u2 + 2as

Convince yourself by deriving these equations !!!

Can you complete the following table?

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What you should understand at the end of this chapter?

s 	= (u + v) * t / 2
 * Motion is a change of position. 	It can be described in terms of the distance moved or displacement.
 * The motion of an object could be 	uniform or non-uniform depending on whether its velocity is constant 	or changing.
 * The speed of an object is the 	distance covered per unit time and velocity is the displacement per 	unit time
 * The acceleration of an object is 	the change in velocity per unit time.
 * Uniform and non-uniform motions 	of objects can be shown through graphs.
 * The motion of an object moving 	at uniform acceleration can be described with the help of the 	following equations.

a	= (v – u)/ t

s	= ut + ½ at2

s	= vt – ½ at2

v2	= u2 + 2as

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Check your understanding


 * 1) A bus starting from rest moves 	with a uniform acceleration of 0.1 m/s2 for 2 minutes. 	Find the (i) speed acquired and (ii) the distance traveled.
 * 2) A train is traveling at a speed 	of 90 km/h. Brakes are applied so as to produce a uniform 	acceleration of -0.5 m/s2.Find how far the train will 	go before it is brought to rest.
 * 3) A trolley while rolling down an 	inclined plane has an acceleration of 2 cm/s2. What will 	its velocity 3 seconds after the start?
 * 4) A racing car has a uniform 	acceleration of 4 m/s2. What distance will it cover in 	10s after the start?
 * 5) A stone is thrown vertically 	upward with a velocity of 5 m/s. If the acceleration of the stone 	is 10 m/s2 in the downward direction, what is the maximum 	height the stone will go up to? How long will it take to reach that 	height?
 * 6) A driver of a car traveling at 	52 km/h applies the brakes and accelerates uniformly in the opposite 	direction. The car stops in 5 seconds. Another driver going at 3 	km/h in another car applies his brakes slowly and stops in 10 	seconds. On the same graph paper, plot the speed versus time graphs 	for the two cars. Which of the two cars traveled farther when the 	brakes were applied?

[[Media:Example.ogg]]
 * 1) S[[Image:Motion%20in%20One%20Dimension_html_5ab79292.gif]]tudy 	the graph and answer the following questions.
 * 2) * Which of the three is traveling 		the fastest?
 * 3) * Are all three ever at the same 		point on the road?
 * 4) * How far has C traveled when B 		passes A?
 * 5) * How far has B traveled by the 		time it passes C?
 * 6) The speed time graph of a body 	is given.
 * 1) The speed time graph of a body 	is given.
 * F[[Image:Motion%20in%20One%20Dimension_html_1ede4a47.gif]]ind 	out how far the car travels in the first 4 seconds. Shade the area 	on the graph that represents the distance traveled by the car during 	the period.
 * Which part of the graph 	represents uniform motion of the car?