In everyday life, some objects are at rest while others are in motion.
Motion is perceived when an object's position changes with time.
Sometimes motion is inferred through indirect evidence, like the movement of air observed through dust or leaves.
The phenomena of sunrise, sunset, and changing seasons are due to the Earth's motion, which we don't directly perceive.
An object's motion is relative; it may appear to be moving to one person and stationary to another.
Most motions are complex, involving straight lines, circular paths, rotation, or vibration, or a combination of these.
The chapter focuses on describing motion along a straight line using equations and graphs, and later discusses circular motion.
Describing Motion
The location of an object is described by specifying a reference point called the origin.
For example, a school is 2 km north of the railway station, where the railway station is the reference point.
Motion Along a Straight Line
The simplest type of motion is motion along a straight line.
Consider an object moving along a straight path, starting from point O (the reference point).
Points A, B, and C represent the object's position at different times.
The object moves from O to A, then back to C through B.
Distance: The total path length covered by the object.
In the example, the distance covered is OA + AC = 60 km + 35 km = 95 km.
Distance is described by specifying only the numerical value (magnitude).
Displacement: The shortest distance measured from the initial to the final position of an object.
It has both magnitude and direction.
In the example, the displacement from O to C is the shortest distance between O and C.
The magnitude of displacement can be equal to the distance traveled by an object when the object moves along a straight line without changing direction.
For motion from O to A, the distance and magnitude of displacement are both 60 km.
The magnitude of displacement is not equal to the path length when the object changes direction during its motion.
For motion from O to A and back to B, the distance covered is 60 km + 25 km = 85 km, while the magnitude of displacement is 35 km.
The magnitude of displacement for a course of motion may be zero, but the corresponding distance covered is not zero.
If the object travels back to O, the displacement is zero but the distance covered is OA + AO = 60 km + 60 km = 120 km.
Distance and displacement are different physical quantities.
Automobiles use an odometer to show the distance traveled.
Uniform Motion and Non-Uniform Motion
Uniform Motion: An object covers equal distances in equal intervals of time.
Example: An object travels 5 m in each second.
Non-Uniform Motion: An object covers unequal distances in equal intervals of time.
Example: A car moving on a crowded street or a person jogging in a park.
Measuring the Rate of Motion
Different objects may take different amounts of time to cover a given distance.
The rate at which objects move can be different.
One way to measure the rate of motion is to find the distance traveled in unit time, which is called speed.
The SI unit of speed is metre per second (m/s).
Other units of speed include centimetre per second (cm/s) and kilometre per hour (km/h).
The speed of an object need not be constant; in most cases, objects are in non-uniform motion.
Average Speed: The total distance traveled divided by the total time taken.
average \ speed = \frac{Total \ distance \ travelled}{Total \ time \ taken}
If an object travels a distance s in time t, then its speed v is: v = \frac{s}{t}
Example: A car travels 100 km in 2 h. Its average speed is 50 km/h.
Speed with Direction
If we specify the direction of motion along with speed, the quantity is called velocity.
Velocity is the speed of an object moving in a definite direction.
The velocity of an object can be uniform or variable.
Velocity can be changed by changing the object's speed, direction of motion, or both.
Average Velocity: When an object is moving along a straight line at a variable speed, we express the magnitude of its rate of motion in terms of average velocity.
Calculated the same way as average speed if the direction does not change.
If the velocity of the object is changing at a uniform rate, then average velocity is given by the arithmetic mean of initial velocity and final velocity for a given period of time.
average \ velocity = \frac{initial \ velocity + final \ velocity}{2}
Mathematically, v{av} = \frac{u + v}{2}, where v{av} is the average velocity, u is the initial velocity, and v is the final velocity of the object.
Speed and velocity have the same units (m/s).
Rate of Change of Velocity
During uniform motion along a straight line, the velocity remains constant with time.
In non-uniform motion, velocity varies with time.
Acceleration: A measure of the change in the velocity of an object per unit time.
acceleration = \frac{change \ in \ velocity}{time \ taken}
If the velocity of an object changes from an initial value u to the final value v in time t, the acceleration a is: a = \frac{v - u}{t}
This kind of motion is known as accelerated motion.
Acceleration is positive if it is in the direction of velocity and negative when it is opposite to the direction of velocity (deceleration or retardation).
The SI unit of acceleration is m/s^2.
Uniform Acceleration: If an object travels in a straight line and its velocity increases or decreases by equal amounts in equal intervals of time.
Example: The motion of a freely falling body.
Non-Uniform Acceleration: If an object's velocity changes at a non-uniform rate.
Example: A car traveling along a straight road increases its speed by unequal amounts in equal intervals of time.
Graphical Representation of Motion
Graphs provide a convenient method to present basic information about motion.
Distance-Time Graphs
The change in the position of an object with time can be represented on a distance-time graph.
Time is taken along the x-axis, and distance is taken along the y-axis.
Uniform Speed: When an object travels equal distances in equal intervals of time, the distance traveled is directly proportional to time taken.
A graph of distance traveled against time is a straight line.
The distance-time graph can be used to determine the speed of an object.
v = \frac{s2 - s1}{t2 - t1}
Non-Uniform Speed: The distance-time graph is a curve.
Velocity-Time Graphs
The variation in velocity with time for an object moving in a straight line can be represented by a velocity-time graph.
Time is represented along the x-axis, and velocity is represented along the y-axis.
Uniform Velocity: The height of its velocity-time graph will not change with time and is a straight line parallel to the x-axis.
The area enclosed by the velocity-time graph and the time axis will be equal to the magnitude of the displacement.
Uniformly Accelerated Motion: The velocity-time graph is a straight line.
The area under the velocity-time graph gives the distance (magnitude of displacement) moved by the car in a given interval of time.
Non-Uniformly Accelerated Motion: Velocity-time graphs can have any shape.
Equations of Motion
When an object moves along a straight line with uniform acceleration, its velocity, acceleration, and the distance covered can be related by a set of equations known as the equations of motion.
v = u + at
s = ut + \frac{1}{2}at^2
2as = v^2 - u^2
Where:
u is the initial velocity.
v is the final velocity.
a is the uniform acceleration.
t is the time.
s is the distance traveled.
Uniform Circular Motion
When the velocity of an object changes, the object is accelerating.
The change in velocity could be due to change in its magnitude or the direction of the motion or both.
Uniform Circular Motion: When an object moves in a circular path with uniform speed.
The only change in velocity is due to the change in the direction of motion.
The motion is an example of accelerated motion.
The circumference of a circle of radius r is given by 2\pi r.
If the athlete takes t seconds to go once around the circular path of radius r, the speed v is given by:
v = \frac{2 \pi r}{t}
Examples: The motion of the moon and the earth, a satellite in a circular orbit around the earth, a cyclist on a circular track at constant speed.