In everyday life, objects are either at rest or in motion.
Motion is perceived when an object's position changes with time.
Motion can also be inferred through indirect evidence, such as observing the movement of dust or leaves to infer the motion of air.
The apparent motion of an object depends on the observer's frame of reference. For example, roadside trees appear to move backward for passengers in a moving bus, while a person on the roadside sees the bus and passengers moving.
Motions can be complex, including straight-line, circular, rotational, and vibrational movements, 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, known as the origin.
Example: A school is 2 km north of the railway station. The railway station is the reference point.
Motion Along a Straight Line
Simplest type of motion.
To describe motion, consider an object moving along a straight path, starting from point O (the reference point).
Points A, B, and C represent the object's positions 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 (e.g., OA + AC).
In the example, OA + AC = 60 km + 35 km = 95 km.
Requires only the numerical value (magnitude) to be specified.
Displacement: The shortest distance between the initial and final positions of the object (e.g., from O to C).
For the object moving from O to A, the distance covered is 60 km, and the magnitude of displacement is also 60 km.
For the object moving 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 can be zero even if the distance covered is not zero (e.g., if the object returns to its starting point).
Example: If the object travels back to O, the displacement is zero, but the distance covered is OA + AO = 60 km + 60 km = 120 km.
Uniform 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.
Measuring the Rate of Motion
The rate of motion is measured by finding the distance traveled by the object in unit time (speed).
Speed: Distance traveled per unit time.
SI unit: metre per second (m/s or m s^{-1}).
Other units: centimetre per second (cm/s or cm s^{-1}), kilometre per hour (km/h or km h^{-1}).
Requires only magnitude.
Average Speed: 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, its speed v is: v = \frac{s}{t}.
Example: A car travels 100 km in 2 hours. Its average speed is 50 km/h.
Velocity: Speed of an object moving in a definite direction.
Can be uniform or variable.
Can be changed by changing speed, direction, or both.
Average Velocity:
For an object moving along a straight line at variable speed, the magnitude of its rate of motion is expressed as average velocity.
If the velocity of the object is changing at a uniform rate, then the average velocity is given by the arithmetic mean of the 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.
Speed and velocity have the same units (m/s).
Rate of Change of Velocity
During uniform motion along a straight line, velocity remains constant.
In non-uniform motion, velocity varies with time.
Acceleration: 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 a final value v in time t, the acceleration a is: a = \frac{v - u}{t}.
SI unit: m s^{-2}.
Uniform Acceleration: An object travels in a straight line, and its velocity increases or decreases by equal amounts in equal intervals of time.
Example: Motion of a freely falling body.
Non-Uniform Acceleration: An object's velocity changes at a non-uniform rate.
Example: A car increasing its speed by unequal amounts in equal intervals of time.
The acceleration is positive if it is in the direction of velocity and negative when it is opposite to the direction of velocity.
Graphical Representation of Motion
Graphs provide a convenient method to present basic information about events.
Line graphs can be used to describe the motion of an object.
Distance-Time Graphs
The change in the position of an object with time can be represented on the distance-time graph.
Time is taken along the x-axis.
Distance is taken along the y-axis.
Distance-time graphs can be employed under various conditions where objects move with uniform speed, non-uniform speed, remain at rest etc.
For uniform speed, a graph of distance traveled against time is a straight line.
The portion OB of the graph shows that the distance is increasing at a uniform rate.
Speed can be determined from the graph by considering a small part AB. AC denotes the time interval (t2 – t1), while BC corresponds to the distance (s2 – s1). The speed, v of the object is given by: v = \frac{s2 - s1}{t2 - t1}.
For accelerated motion, the distance-time graph is not a straight line, indicating the non-linear variation of distance with time.
The nature of this graph shows non- linear variation of the distance travelled by the car with time.
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.
Velocity is represented along the y-axis.
For an object moving at uniform velocity, the velocity-time graph will be 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.
The distance s moved by the car in time (t2 – t1) can be expressed as: s = AC × CD = [(40 \ km \ h^{-1}) × (t2 – t1) h] = 40 (t2– t1) \ km.
For uniformly accelerated motion, the velocity-time graph is a straight line.
The distance s traveled by the car will be given by the area ABCDE under the velocity-time graph.
s = area \ ABCDE = area \ of \ the \ rectangle \ ABCD + area \ of \ the \ triangle \ ADE = AB × BC + \frac{1}{2} (AD × DE).
In the case of non-uniformly accelerated motion, velocity-time graphs can have any shape.
Velocity-time graph that represents the motion of an object whose velocity is decreasing with time
Velocity-time graph representing the non-uniform variation of velocity of the object with time.
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 the equations of motion:
v = u + at
s = ut + \frac{1}{2} a t^2
2 a s = 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.
Eq. (7.5) describes the velocity-time relation.
Eq. (7.6) represents the position-time relation.
Eq. (7.7) represents the relation between the position and the velocity.
Uniform Circular Motion
When the velocity of an object changes, we say that the object is accelerating.
The change in velocity could be due to change in its magnitude or the direction of the motion or both.
When an object does not change its magnitude of velocity but only its direction of motion, it is in circular motion.
If the athlete moves with a velocity of constant magnitude along the circular path, the only change in his velocity is due to the change in the direction of motion. The motion of the athlete moving along a circular path is, therefore, an example of an 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}
When an object moves in a circular path with uniform speed, its motion is called uniform circular motion.