Motion
Category : 9th Class
Motion
Chapter Overview
The branch of physics which deals with the study of motion of objects is called mechanics.
Statics is the study of object at rest and Dynamics is the study of object in motion.
The study of dynamics is further subdivided into:
(a) Kinematics
(b) Dynamics Proper.
Kinematics is the study of motion without taking into account the cause of motion.
Dynamics proper is the study of motion taking into account the cause of motion.
A body is said to be rest if it does not change its position with respect to its immediate surroundings.
On the other hand, a body is said to be in motion if it changes its position with respect to its immediate surroundings.
Example 1: When a tree T, is observed by m observer A sitting on a bench, the tree is at rest. This is because the position of the tree is not changing with respect to the observer A.
Now, when the same tree T is observed by an observe r B sitting in a superfast train moving with a velocity v, then the tree is moving with respect to the observer because the position of tree is changing with respect to the observer B.
2. Types of Motion
(1) Linear Motion: An object has linear motion if it moves in a straight line or path. It is also called rectilinear motion.
Examples of linear motion:
(i) Motion of a moving car on a straight road.
(ii) Motion of a ball dropped from the roof of a building.
(2) Circular (or Rotatory Motion): An object has circular motion if it moves around a fixed point.
The blades of a fan rotates around a fixed point and therefore have rotatory motion.
(3) Vibratory Motion: A body has vibratorynotion if it moves to and fro about a fixed point.
This type of motion called Vibratory Motion.
Example - The motion of a sitar string when plucked exhibits Vibratory Motion.
Scalar Quantity The physical quantity which h ave only nagnitude an d no sense of direction is called scalar quantity
Examples: Mass, time, distance, speed, work, power, volume, densityetc.
A magnitude is represented by a number and a unit.
Remember: Scalars obey normal algebraic operations.
Vector quantity A physical quantity which is described completely its magnitude as well as its direction is called a vector quantity
If an object is moving with a speed of 40 km/h in a particular direction, say south. We say that the velocity the car is 40 km/h due south.
Example 1: Displacement, velocity torque, acceleration, fore e, momentum etc.
Remember: Vectors do not obey normal algebraic operation.
Distance: The actual length of path covered by moving body between its initial and final position is called distance covered by the body
Displacement: When a body moves from one position to another, the shortest distance (straight distance) measured between initial and the final positions of the moving body in a particular direction is called displacement.
Displacement: When a body moves from one position to another, the shortest distance (straight distance) measured between initial and the final positions of the moving body in a particular direction is called displacement.
Note- Displacement of an object = (Final position-Initial position) of the object.
Distance = Displacement
Remember:\[\frac{\text{Distance}}{\text{ }\!\!|\!\!\text{ Displacement }\!\!|\!\!\text{ }}\ge 1\]
Difference between Distance and Displacement
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Parameter |
Distance |
Displacement |
|
1. Definition |
The actual length of the path travelled by a moving body irrespective of the direction is called the distance travelled by that body. |
The shortest distance measured between the initial and the final position of a moving body in a particular direction is called its displacement. |
|
2. Physical quantity |
It is a scalar quantity. |
It is a vector quantity. |
|
3. Value |
It is always positive (It can be zero or negative). |
It may be positive, negative or zero. |
|
4. Dependence on Path |
It depends on the path followed by the moving object. |
Its does not depend on the path followed by the moving object. |
|
5. Magnitude |
Its magnitude is always greater than or equal to the displacement. |
Its magnitude is always less than or equal to the distance travelled. |
An expert controller sitting in a control room monitors how fast various trains are moving and where exactly each one of them is located at a given instant of time so that he can give correct signals and prevent train accidents.
To describe motion we need to classify it as uniform motion or non-uniform motion.
Uniform Motion: A body is said to have uniform motion if it travels equal distances in equal intervals of time, no matter how small these time intervals may be.
Fig. 5.1
Examples of Uniform motion:
(1) The movement of hands of watches.
(2) The movement of the earth about its axis.
(3) The movement of the earth around the sum.
(4) A gas molecule is in uniform motion between collisions.
Non-Uniform Motion: A body is said to have non-uniform motion if it travels unequal distances in equal intervals of time, however small these, time intervals may be.
Examples of non-uniform motion:
(1) A bus moving on a hilly track.
(2) A person jogging on a road.
(3) A free-falling stone under the action of gravity.
(4) When brakes are applied to a speeding car.
(5) When an oscillating simple pendulum is left for some time, the amplitude of its oscillation becomes smaller and smaller and finally the oscillation stops.
The distance time graph of a body having non uniform motion is a curved line.
Fig. 5.2
To compare motion of two or more objects moving in any direction one must have an understanding of the concept of speed and velocity.
(i) Speed: Speed of a body is the distance travelled by the body per unit time.
\[\text{Speed}=\frac{\text{Distance travelld}}{\text{Time}\,\text{taken}}\]
Speed is denoted by y. The distance travelled in an interval of time t is denoted by s.
\[\text{v}=\frac{s}{t}\]
To specify the speed of a moving object, we require only its magnitude (direction is not described) so, speed is a scalar quantity,
The S.I. unit of speed is m/s (or\[\text{m}{{\text{s}}^{-1}}\]) and C.G.S. unit cm/s (or\[\text{cm}{{\text{s}}^{-1}}\]).
Another unit of speed is km/h or\[\text{km}{{\text{s}}^{-1}}\].
Changing the unit: \[1\frac{km}{h}=\frac{1000m}{60\times 60s}=\frac{5m}{18s}\left( \text{km/h to m/s} \right)\]
\[\Rightarrow \]\[1\,\text{m/s}=\frac{18}{5}\text{km/h}\]or\[3.6\,\text{km}{{\text{h}}^{\text{-1}}}\](m/s to km/h)
(ii) Average speed: While travelling in a car we have noticed that the driver changes the peed of the moving car from time to time depending upon the traffic and road conditions.
So, we describe the rate of motion of such objects in terms of their average speed.
The average speed of a body in a given time interval is defined as the total distance travelled divided by the time interval,
\[\text{Average}\,\text{speed=}\frac{\text{Total distance travelled}}{\text{Total}\,\text{time}\,\text{taken}}\]
(iii) Instantaneous speed: The speed of an object at any instant during its motion is called Instantaneous speed.
Automobiles are fitted with a device called an odometer which measures the distance travelled by it.
The odometer records the distance in kilometers.
Remember: When a body moves with uniform speed, then the average speed is same as instantaneous speed. Speedometer measures the instantaneous speed of vehicle.
(iv) Velocity: Velocity of a body is the displacement of a body per unit time.
\[\text{Velocity=}\frac{\text{Displacement}}{\text{Total}\,\text{interval}}\]
The direction of velocity is same as the direction of displacement.
Velocity has both magnitude and direction, so velocity is vector quantity.
Expression for velocity: If the displacement of the body is s in time \[\vec{s}\]then velocity of the body is given by
\[\text{\vec{v}=}\frac{\text{Displacement}}{\text{Time}}=\frac{{\vec{s}}}{t}\]
S.I. unit of velocity is metre/second \[(m{{s}^{-1}})\].
In C.G.S. system, unit of velocity is centimeter/second \[(cm{{s}^{-1}})\]
Commonly used unit of velocity is kilometer/hour\[(km{{h}^{-1}})\].
Consider a car moving around a circular track at a constant speed of 50 km/h. At every point on the track the speed is same i.e., 50 km/h.
But at every point (such as A, B and C) the direction of moving car continuously changes along the circular track. Thus, the magnitude of velocity at points A, B and C is the same but the direction of motion is changing, so we say that the body is having a variable velocity.
A body is said to have variable velocity if it covers unequal distances in equal distances m equal interval of time m a particular direction.
Fig.:7.1.
OR
A body is said to have variable velocity if it covers equal distances in equal interval of time but its direction keeps on changing.
Examples of non-uniform or variable velocity:
(i) A car moving towards south on crowded road.
(ii) A stone dropped from the top of a building.
(iii) A body moving in a circular path for example, a moving fan.
Average velocity: When the velocity of a moving body is changing continuously at a uniform rate, then the average velocity is given by the arithmetic mean of initial and final velocities for a given period of time.
\[\text{Average}\,\text{velocity=}\frac{\text{Initial velocity + Final velocity}}{2}\]
Mathematically, \[{{\text{u}}_{a\text{v}.}}\text{=}\frac{\text{u+v}}{2}\]
where \[{{\text{u}}_{a\text{v}.}}\text{=}\]average velocity
u = initial velocity of the moving body
v = final velocity of the moving body.
Also the magnitude of velocity should remain constant.
In such case since distance = displacement, we can use the word speed also for referring to the magnitude of velocity.
A body has variable velocity when:
(a) Its magnitude changes but direction remain the same. e.g,, a train coming to station platform on a straight track.
(b) Its magnitude does change but the direction of motion changes.
e.g., when a bus takes a turn with its speed remaining constant.
(c) Its magnitude as well as the direction of motion changes.
Instantaneous Velocity: Instantaneous velocity of a body is the velocity of a body at an instant of time.
\[\text{Velocity=}\frac{\text{Displacement}}{\text{Time}\,\text{Interval}}\]
Comparison between Speed and Velocity
|
|
Remember |
Speed |
Velocity |
|
1. |
Definition |
The speed of a body is the distance travelled by it per unit time. |
The velocity of a body is the distance travelled by it per unit time in a definite direction. |
|
2. |
Physical quantity |
It is a scalar quantity |
It is a vector quantity. |
|
3. |
Value |
It is always positive (It can be zero or negative) |
It can be positive, negative or zero. |
|
4. |
Formula |
|
|
The change in velocity of an object with time is expressed by a physical quantity known as acceleration.
Definition: The rate of change of velocity of a body with respect to time is called its acceleration.
Acceleration of a body or an object is defined as the change in velocity per unit time.
\[\text{Acceleration=}\frac{\text{Change}\,\text{in}\,\text{velocity}}{\text{Time}\,\text{taken}\,\text{to}\,\text{change}}\]
Change in velocity = Final velocity – Initial velocity
\[\text{Acceleration=}\frac{\text{Final}\,\text{velocity-Initial}\,\text{velocity}}{\text{Time}\,\text{taken}\,\text{to}\,\text{change}}\]
Initial velocity of the body = u
Final velocity of the body = v
Time taken to change = t
Acceleration (a)\[\text{=}\frac{u-u}{t}\]
Units of Acceleration:
In the S.I. system: \[m/{{s}^{2}}\] or \[m{{s}^{-2}}\]
In the C.G.S. system: \[cm/{{s}^{2}}\] or \[cm{{s}^{-}}^{2}\]
The other unit of acceleration is kilometer per hour square written as \[km{{h}^{-}}^{2}\]or \[km/{{h}^{2}}\].
Acceleration is a vector quantity:
(i) Positive Acceleration: When the velocity of a body increases with time its acceleration is positive.
Acceleration of an object is positive if its direction is same as that of the direction of motion of the object.
Example: If an object starts from rest and its velocity goes on increasing with time in the direction of its motion, then the object has positive acceleration.
In this case, the final velocity of moving body is greater than the initial velocity, i.e., v > u.
Example: a body falling towards the earth has positive acceleration.
(ii) Negative Acceleration: When the velocity of a body decreases with time its acceleration is negative. Negative acceleration is also called retardation or deceleration.
In this case, the final velocity of the moving body is less than the initial velocity v < u.
Acceleration (a) \[=\frac{\text{v}-u}{t}=\]negative quantity, since v < u
=- a
Negative sign indicates that the direction of acceleration is opposite to the direction of motion of an object.
10. Difference between Acceleration and Retardation
|
|
Parameter |
Acceleration |
Retardation |
|
1. |
Definition |
If the velocity of a body increases, then the rate of change of velocity is positive and is called acceleration. |
It the velocity of a body decreases then the rate of change of velocity is negative and is called retardation. |
|
2. |
Example |
A body falling freely from a certain height. |
If an object is thrown in the upward direction its velocity decreases. |
|
3. |
Relation |
u > u |
u > v |
If the velocity of an object changes by an equal amount in equal intervals of time, then the acceleration of the object is known as uniform acceleration.
The motion of body with uniform acceleration is called uniformly accelerated motion.
Example: (1) A body falling freely under gravity has uniform acceleration.
(2) An object moving down on inclined plane has uniform acceleration.
(b) Non-uniform Acceleration (Variable Acceleration): When the velocity of a body changes by unequal amount in equal intervals of time, it is said to have non-uniform acceleration.
The motion of an object having variable acceleration is known as non uniformly accelerated motion.
Examples of non-uniformly accelerated motion:
(i) The motion of a train leaving or entering the platform.
(ii) The motion of a bus on a crowded road.
A graph is a line, straight or curved, showing the relation between two variable quantities (or their powers of functions) of which one varies as a result Of the change in the other.
The quantity that is alter at self-choice, is called independent variable and the other which varies as a result of this change is called the dependent variable.
Thus, the relation between natural numbers (independent variable) and their squares (dependent variable) can be shown by means of a graph.
Remember: A graph not only gives the relation between two variable quantities in a pictorial form but also enable (i) To study nature of motion,
(ii) To verify Newton's law of motion.
The variation of the dependent quantity with an independent quantity is represented by a graph known as line graph.
Here we will study the following graphs:
While drawing the graphs, time is always taken along the X-axis.
(1) Distance-time Graph: A graph-showing the change in the position of an object (distance covered) with the change in time is called its distance-time graph.
To draw distance-time graph distance travelled by the body is plotted along Y-axis and the time taken by the body to travel this distance is plotted along X-axis.
We can study the distance-time graph of a body under the following conditions:
(a) When the body is moving with a uniform speed,
(b) When the body is moving with a non-uniform speed.
(c) When the body is at rest.
(i) Stance-time graph for a Stationary Body: The distance of a stationary body from fixed point does not change with the passage of time.
Fig.: 12.1
If the distance-time graph of a body is a straight line parallel to the X-axis the body is at rest.
\[\text{Speed}\,\text{of}\,\text{an}\,\text{object=}\frac{\text{Distance}\,\text{covered}}{\text{Time}\,\text{taken}}\]
\[\text{v=}\frac{0}{{{t}_{2}}-{{t}_{1}}}=0\]
The speed of the body at rest is zero.
Conclusion: Distance-time graph for a stationary body is a straight line parallel to time axis Distance-time Graph for Uniform Motion: If a body travels equal distances m equal intervals of time then the motion of the body is known as uniform motion.
The distance-time graph of a body is a straight line, the body is moving with uniform speed.
Slope is the ratio of change in physical quantity represented on the vertical axis (say Y-axis) to the change in physical quantity represented on the horizontal axis (say X-axis).
Fig.: 12.2.
Conclusion: Distance time graph for the uniform motion of an object having a constant angle with the time axis is of constant gradient or slope.
Use of distance-time graph for uniform motion:
Calculation of the speed of a body:
\[\text{Speed=}\frac{\text{Distance travelled}}{\text{Time taken to travel this distance}}\]
The distance travelled by the body is going from A to B.
\[\Delta r=BC-CD\]
\[={{s}_{2}}-{{s}_{1}}\]
= 10 - 4 = 6 m
Time taken by the body to cover this distance\[=\Delta t\]
\[={{t}_{2}}-{{t}_{1}}\]
= 5 – 2 = 3 sec.
Speed \[=\frac{\Delta x}{\Delta t}=\frac{{{s}_{2}}-{{s}_{1}}}{{{t}_{2}}-{{t}_{1}}}=\frac{6}{3}=2\,m{{s}^{-1}}\]
Here \[\frac{\Delta x}{\Delta t}=\]slope of distance-time graph
Fig.: 12.3.
Remember: speed of a body= Slope of distance-time graph in uniform motion.
Slope of a line tells us how steeply it is inclined to the X-axis. As the line gets more & more inclined to the X-axis it makes a larger angle with X-axis (time axis) i.e., its slope increases.
A more steeply inclined distance-time graph indicates greater speed.
In the figure shown above the straight line for object A makes a larger angle with the time axis. Its slope is larger than the slope of the line of object B. So, the speed of A is greater than that of B.
(iii) Distance-Time Graph for Non Uniform Motion: If a body travels unequal distances in equal interval of time, then the motion of the body is known as non-uniform motion.
Non-uniform motion of a body is of two types:
So therefore, the bus covers more and more distance in unit time.
Conclusion: Distance time graph for the non-uniform motion of a body is a curve having increasing slope or gradient. The speed (or velocity) of such a body changes with time and hence the motion of the body is accelerated.
2. When the speed of the body decreases with passage of time: When a bus approaches the stoppage, its speed decreases with passage of time. That is, the bus covers less and less distance in unit time.
Fig: 12.4.
Important Points
Conclusion: Distance-time graph for the decreasing non-uniform motion of a body is a curve having decreasing slope.
Speed of the body decreases with passage of time.
[Note: (1) Displacement-time graph is similar to the distance-time graph, if the object moves in a straight line along one direction.]
(2) Velocity-time graph: A graph showing the variation of velocity with the change in time is called a velocity-time graph.]
In this graph, time t is taken along X-axis and the velocity v is taken along Y-axis. We study the velocity time graph of a body under following conditions:
Fig.: 12.5.
(a) When the body is moving with a constant velocity.
(b) When the body is moving with uniformly accelerated motion (i.e., velocity increases uniformly with time).
(c) When the body is moving with uniformly decelerated motion (velocity decreases uniformly with time).
(d) When the body is moving with non-uniformly accelerated motion (i.e., velocity increases or decreases non-uniformly with time).
(A) Velocity-time Graph for a body moving with constant velocity: When an object moves with constant velocity i.e., its motion is uniform, its velocity does not change with time.
Fig. 12.6.
Conclusion: Velocity time graph of a body moving with constant velocity is a straight line parallel to time axis.
Use of velocity -time graph for a body moving withoastant velocity in calculation of magnitude of displacement:
Magnitude of displacement of body = area under velocity-time graph
= Area of rectangle ABCD
\[=DA\times CD\]
\[=6\times 3=18m\]
Note- Speed-time graph for a body moving with constant speed is same as shown in fig.
The only difference is that speed is written instead of velocity along Y-axis.
Fig.: 12.7
Distance travelled by a body = area under speed-time graph
= area of rectangle OABC
\[=OA\times OC\].
(B) For uniform motion when initially the body is at rest.
Initially when the object is at rest the velocity of body at t = 0.
In every unit interval of time, the velocity of the body increases by an equal amount.
Fig.: 12.8.
Velocity time graph of a body having uniform motion (initially at rest) is a straight line passing through the origin and have a constant slope (or constant gradient).
With the help of (u -t) graph calculate the acceleration of an object:
\[\text{Acceleration}\frac{\text{Change}\,\text{in}\,\text{velocity}}{\text{Time}\,\text{taken}}\]
\[=\frac{AB}{OB}=\frac{\Delta v}{\Delta t}=\frac{(10-0)m{{s}^{-1}}}{(5-0)s}=2\,m{{s}^{-2}}\].
Calculation of magnitude of displacement:
Magnitude of displacement = Area under u -1 graph
= Area under OA
= Area of triangle OAB
\[=\frac{1}{2}\times base\times height\]
\[=\frac{1}{2}OB\times AB\]
\[=\frac{1}{2}\times 5s\times 10m{{s}^{-1}}\]
= 25 m.
(C) Velocity-time graph when the velocity of the body decreases uniformly to zero: Let the velocity of the bus at t = 0 is \[10\text{ }m{{s}^{-}}^{1}\]. The driver of the bus applies brakes so that the velocity of the train decreases uniformly to zero.
Fig.: 12.9.
Uses: (i) Calculation of acceleration,
Acceleration = slope of u - t graph
= slope of AB line
\[=\frac{OA}{OB}=\frac{-10m{{s}^{-1}}}{5\sec .}=-2m{{s}^{-2}}\]
Negative sign shows that acceleration of the train is negative.
(ii) Displacement of body,
Displacement = area of \[\Delta AOB\]
\[=\frac{1}{2}\times base\times height=\frac{1}{2}\times ({{t}_{2}}-{{t}_{1}})({{v}_{2}}-{{\text{v}}_{1}})\]
\[=\frac{1}{2}\times 5m\times 10m{{s}^{-1}}=25m\]
(D) Velocity-time graph for non-uniformly accelerated motion:
(i) When the velocity increases non-uniformly the v -1 graph is a curve moving upwards as shown in the graph given below:
Fig.: 12.10.
(ii) When the velocity decreases non-uniformly. Velocity-time graph is a curve moving downwards as shown in the graph given below:
Fig.: 12.11.
The uniformly accelerated motion of a body is described by three equations known as equations of motion.
(i) First equation of motion: Velocity-time relation.
v = u + at
Second equation of motion: Position-time relation
\[s=ut+\frac{1}{2}a{{t}^{2}}\]
Third Equation of motion: Position-velocity relation.
\[{{v}^{2}}={{u}^{2}}+2as.\]
The motion of a body moving around a fixed point in a circular path is known as circular motion.
Examples of circular motion:
(1) The motion of a satellite around the earth.
(2) The motion of moon around the earth.
(3) A wheel rotating about its axle.
(4) The motion of an electron around the nucleus of an atom.
Uniform Circular Motion: When a point object (small body or particle) is moving on a circular path with a constant speed, the motion of the object is said to be uniform circular motion.
A stone tied to a thread and whirled in a circular path is an example of uniform circular motion.
\[v=\frac{2\pi r}{t}\]
For a body moving with a constant speed along a circular path, the direction of the velocity is along the tangent to the circle at any point in its motion.
Fig.: 14.1.
Misconception: When a body moves with constant speed, its acceleration is zero.
Conception: When a body moves with constant speed but its direction of motion changes, then the velocity changes and the body is accelerating. Such motion is called accelerated motion.
Chapter at a Glance
\[\text{Speed=}\frac{\text{Distance}}{\text{Time}}\]
State of object:
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1. Rest or stationary
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2. Uniform motion
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3. Non-uniform motion (speed is increasing)
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4. Non-uniform motion (speed is decreasing)
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Fig: 14.2. |
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State of object:
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1. Object is moving with constant velocity.
|
2. Object has uniform motion (velocity is increasing at a constant rate).
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3. Object has non-uniform motion (velocity is decreasing at a constant rate).
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4. Object is moving with non-uniform velocity.
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Fig.:12.3. |
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(i) Velocity of a uniformly accelerated body after time t is given by
v = u + at.
(ii) Distance travelled by a uniformly accelerated body after time t is given by
\[s=ut+\frac{1}{2}a{{t}^{2}}\].
(iii) Distance travelled by a body during the period with velocity changes from u to u.
\[{{v}^{2}}-{{u}^{2}}=2as\]
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