2. Motion in One Dimension

Motion is the change in position of a body with respect to time and a reference point.

Motion refers to the change in the position of an object as time passes. To understand motion, it's important to reference a point, known as a reference point, which helps to determine if an object is moving or stationary. For example, when you are on a moving bus, everything outside the bus is your reference point, and how the bus changes its position regarding that reference is what we describe as motion.

Imagine you are watching a train pass. The trees and buildings outside remain stationary, while the train moves from one point to another. This illustrates motion with respect to a stationary background.

● Translatory motion: Linear movement (straight line or curve) ● Rotational motion: About a fixed axis ● Vibratory motion: To and fro about a mean position

There are various types of motion which can be classified as follows:

1. Translatory motion refers to the movement along a straight path or a curve, like a car driving on a road.
2. Rotational motion involves objects turning around a fixed point or axis, like the Earth rotating around its axis.
3. Vibratory motion consists of back-and-forth movements around a central point, such as a pendulum swinging in a clock.

Think of a merry-go-round. As it spins, riders experience rotational motion. When swinging on a swing, your motion is vibratory, moving back and forth from a central position.

Quantity Description SI Unit Distance Total path length travelled Metre (m) Displacement Shortest distance from initial to final point Metre (m) Speed Rate of change of distance m/s or km/h Velocity Rate of change of displacement m/s or km/h Acceleration Rate of change of velocity m/s²

This section defines essential terms associated with motion.

• Distance is the total length of the path traveled by an object, always a positive quantity measured in meters.
• Displacement is the straight line distance from the starting point to the endpoint and can be positive, negative, or zero depending on direction.
• Speed is how fast an object moves and is calculated as the distance traveled over time, measured in meters per second (m/s) or kilometers per hour (km/h).
• Velocity, similar to speed, also considers direction and is the displacement over time.
• Acceleration measures how quickly an object’s velocity changes and is expressed as meters per second squared (m/s²).

If you drive 3 km east and then 4 km west, your distance is 7 km, but your displacement is 1 km west. If you take ten seconds to cover that distance, your speed is 0.7 km/h, while the velocity accounts for the direction and could be negative if we define east as positive.

Feature Distance Displacement Scalar/Vecto Scalar Vector r Value Always positive Can be positive, negative, or zero Path Actual path Shortest path

This comparison distinguishes distance from displacement.

• Distance is a scalar quantity, meaning it only has magnitude and is always positive.
• Displacement, by contrast, is a vector quantity, which includes both magnitude and direction and can be positive, negative, or zero depending on the object’s motion.
• Also, distance refers to the actual path taken, while displacement refers to the shortest path between two points.

Consider a situation in a park where you walk around a pond taking 100 meters of path. Your distance is 100 meters. However, if the pond was 30 meters across, your displacement (straight line from start to end) would only be 30 meters, showcasing the difference between these two concepts.

Speed = Distance travelled / Time taken
Velocity = Displacement / Time taken
● Uniform speed/velocity: Same magnitude at all times
● Variable speed/velocity: Changes with time
● Average speed = Total distance / Total time
Average velocity = Total displacement / Total time

In understanding speed and velocity:

• Speed is the total distance an object travels divided by the time it takes, while velocity includes directional information and is the ratio of displacement over time.
• If a car travels at the same speed throughout its journey, it has uniform speed, whereas if it speeds up or slows down, it has variable speed.
• Average speed and average velocity provide a general idea of motion over a journey by totaling the distances or displacements covered during the total time of travel.

Imagine driving to a friend's house. Initially, you might go slowly in traffic (low speed) and then speed up on a highway (high speed). Your overall average speed helps determine how long the trip took, but velocity reflects how quickly you reached your destination directly.

Acceleration (a) = Change in velocity / Time taken = (v - u) / t
Where: ● u = initial velocity ● v = final velocity ● t = time taken
● Positive acceleration → speeding up
● Negative acceleration (deceleration) → slowing down

Acceleration describes how quickly an object's velocity changes over time. Its formula indicates the difference between final velocity (v) and initial velocity (u) divided by the time (t) taken.

• If an object speeds up, it has positive acceleration.
• Conversely, if it slows down, it experiences negative acceleration, often called deceleration.

Picture a bicycle racer. As the biker pedals harder going downhill, their speed increases, demonstrating positive acceleration. However, when they tap the brakes to slow down at a turn, that's an example of negative acceleration.

1. Distance-Time Graph
   ● Uniform motion: Straight line with constant slope
   ● Non-uniform motion: Curved line
2. Velocity-Time Graph
   ● Uniform acceleration: Straight sloping line
   ● Area under graph = distance travelled
   ● Slope = acceleration

Graphical representations help visualize motion:

1. Distance-Time Graphs show how distance changes over time. Straight lines indicate uniform motion, while curved lines imply changing speeds.
2. Velocity-Time Graphs depict how velocity varies with time. A straight line suggests uniform acceleration, and the area beneath the line represents the total distance traveled while acceleration represents the slope of the line.

Imagine a race track; a distance-time graph for a runner might show them moving steadily (straight line). If they sprint (acceleration), the graph curves upwards. Conversely, in a velocity-time graph indicating acceleration, the steeper the line goes up, the faster the runner is gaining speed.

1. v = u + at
2. s = ut + 1/2 at²
3. v² = u² + 2as
   Where: ● s = displacement ● u = initial velocity ● v = final velocity ● a = acceleration ● t = time taken

These equations relate initial velocity (u), final velocity (v), acceleration (a), time (t), and displacement (s) in uniformly accelerated motion.

1. The first equation calculates the final velocity after a specific time based on initial velocity and acceleration.
2. The second equation helps determine displacement over time given initial velocity and acceleration.
3. The third enables the calculation of final velocity based on displacement and acceleration.

Think of a car speeding up from a standstill. Using these equations, you can calculate how fast it will be going after a given time or how far it will travel under constant acceleration.

● Free fall: Motion of an object under gravity alone.
● Acceleration due to gravity (g): g ≈ 9.8 m/s²
● In free fall: a = g, u = 0
● Equations used: v = gt, s = 1/2 gt², v² = 2gs

Free fall describes how objects fall toward the Earth solely due to the pull of gravity, with no air resistance affecting them. The acceleration due to gravity is approximately 9.8 m/s². When an object falls without any initial velocity, its motion is described using formulas that show how its speed and distance change over time.

Consider dropping a ball from a height. Without any external forces, it will accelerate downwards at 9.8 m/s² due to gravity. You can use the mentioned equations to predict how fast the ball will be traveling just before hitting the ground and how far it will fall in a given time.

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