Comprehensive theory, key formulas, diagrams, and memory aids for Motion.
In everyday life, we see some objects at rest and others in motion. Birds fly, fish swim, blood flows through veins and arteries, and cars move. Atoms, molecules, planets, stars, and galaxies are all in motion. We often perceive an object to be in motion when its position changes with time.
However, motion is relative. A tree appears stationary to a person standing on the ground, but to a person in a moving train, the tree appears to be moving backward. To a person in the train, their fellow passengers appear to be at rest, even though the train is moving. In this chapter, we shall learn to describe motion along a straight line using concepts of distance, displacement, speed, velocity, and acceleration.
To describe the position of an object, we need to specify a reference point called the origin. For example, if we say a school is 2 km north of the railway station, the railway station is the origin.
Consider a car moving along a straight line from point O (origin) to point C, then to B, and finally to A. Then it reverses and moves back to point C. * Distance: The total path length covered by the object. If the car went O -> A -> C, the distance is the length $OA + AC$. Distance is a scalar quantity (it has only magnitude, no direction). It cannot be zero if the object has moved. * Displacement: The shortest distance measured from the initial to the final position of an object. Displacement is a vector quantity (it has both magnitude and direction). If an object returns to its starting point, its displacement is zero, even though it has covered a certain distance.
Different objects may take different amounts of time to cover a given distance. The rate at which objects move can be different.
One way of measuring the rate of motion of an object is to find out the distance travelled by the object in unit time. This quantity is called speed. * Formula: $\text{Speed} = \frac{\text{Distance}}{\text{Time}} \quad (v = \frac{s}{t})$ * SI Unit: metre per second (m/s or $m \cdot s^{-1}$). * Speed is a scalar quantity. In most practical cases, speed is not constant, so we calculate average speed: $$\text{Average Speed} = \frac{\text{Total Distance Travelled}}{\text{Total Time Taken}}$$
The rate of motion of an object can be more comprehensive if we specify its direction of motion along with its speed. The quantity that specifies both these aspects is called velocity. * Velocity is the speed of an object moving in a definite direction. It is a vector quantity. * Velocity can be changed by changing the object's speed, direction of motion, or both. * If velocity changes uniformly, then: $$\text{Average Velocity} = \frac{\text{Initial Velocity} + \text{Final Velocity}}{2} \quad (v_{av} = \frac{u + v}{2})$$ * The SI unit of velocity is also m/s.
During uniform motion along a straight line, the velocity remains constant with time. In non-uniform motion, velocity varies with time. The physical quantity that measures the change in velocity per unit time is called acceleration.
Graphs provide a convenient method to present basic information about a variety of events.
The change in the position of an object with time can be represented on a distance-time graph (Time on x-axis, Distance on y-axis). * Uniform Motion: The graph is a straight line passing through the origin. The slope of a distance-time graph gives the speed of the object. * Non-uniform Motion: The graph is a curved line, indicating varying speed. * Object at Rest: The graph is a horizontal line parallel to the time axis.
The variation in velocity with time is represented by a velocity-time graph (Time on x-axis, Velocity on y-axis). * Uniform Motion (Constant Velocity): The graph is a horizontal straight line parallel to the time axis. The acceleration is zero. * Uniformly Accelerated Motion: The graph is a straight line inclined to the time axis. The slope of a velocity-time graph gives the acceleration. * Area Under the Curve: The area enclosed by the velocity-time graph and the time axis gives the magnitude of the displacement (distance travelled in a given direction).
When an object moves along a straight line with uniform acceleration, it is possible to relate its velocity, acceleration, and the distance covered by it in a certain time interval using three equations of motion:
(Where $u$ = initial velocity, $v$ = final velocity, $a$ = uniform acceleration, $t$ = time, and $s$ = distance/displacement).
When the velocity of an object changes, we say that the object is accelerating. A change in velocity could be due to a change in its magnitude or the direction of the motion. If an object moves in a circular path with uniform speed, its motion is called uniform circular motion. * Even though the speed is constant, the direction is changing continuously at every point. * Because the direction is changing, the velocity is changing. * Therefore, uniform circular motion is an example of accelerated motion. (e.g., motion of the moon around the earth, a cyclist on a circular track).
Understanding motion requires establishing a frame of reference. We distinguish between scalar quantities like distance and speed, and vector quantities like displacement and velocity, which include direction. Acceleration measures how quickly velocity changes. These concepts can be visualized clearly using distance-time and velocity-time graphs. For objects undergoing uniform acceleration, the three equations of motion provide a powerful mathematical tool to solve real-world problems regarding predicting the object's future position or speed. Finally, motion isn't restricted to straight lines; uniform circular motion demonstrates that changing direction continuously, even at a constant speed, results in continuous acceleration.