Answer :
Answer:
A vector is a mathematical quantity that has both magnitude (size or length) and direction. It is often represented by an arrow in diagrams, where the length of the arrow represents the magnitude and the direction of the arrow indicates the direction of the vector quantity.
### Motion in a Straight Line
Motion in a straight line is a fundamental concept in physics that deals with the movement of an object along a straight path, typically represented on a coordinate axis. Here’s a detailed explanation of key concepts involved:
1. **Position and Displacement:**
- **Position (s):** It refers to the location of an object along the straight line at a specific instant. It is usually measured from a reference point (origin) on the line.
- **Displacement (\(\Delta s\)):** It is the change in position of an object from one point to another. It is a vector quantity, as it has both magnitude (distance) and direction (from initial to final position).
2. **Velocity:**
- **Average Velocity (\(v_{\text{avg}}\)):** It is defined as the change in position (\(\Delta s\)) divided by the time interval (\(\Delta t\)) during which the change occurred.
\[ v_{\text{avg}} = \frac{\Delta s}{\Delta t} \]
- **Instantaneous Velocity (\(v\)):** It is the velocity of an object at a specific instant in time. It is represented as the limit of the average velocity as the time interval approaches zero.
\[ v = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t} = \frac{ds}{dt} \]
Velocity is a vector quantity because it has both magnitude (speed) and direction (along the straight line).
3. **Acceleration:**
- **Average Acceleration (\(a_{\text{avg}}\)):** It is defined as the change in velocity (\(\Delta v\)) divided by the time interval (\(\Delta t\)).
\[ a_{\text{avg}} = \frac{\Delta v}{\Delta t} \]
- **Instantaneous Acceleration (\(a\)):** It is the acceleration of an object at a specific instant in time. It is represented as the limit of the average acceleration as the time interval approaches zero.
\[ a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt} \]
Acceleration is also a vector quantity, as it has both magnitude (rate of change of velocity) and direction (along the straight line).
4. **Equations of Motion:**
- **First Equation of Motion (for constant acceleration):**
\[ v = u + at \]
Where:
- \( v \) is the final velocity,
- \( u \) is the initial velocity,
- \( a \) is the constant acceleration,
- \( t \) is the time elapsed.
- **Second Equation of Motion:**
\[ s = ut + \frac{1}{2}at^2 \]
Where:
- \( s \) is the displacement,
- \( u \) is the initial velocity,
- \( a \) is the constant acceleration,
- \( t \) is the time elapsed.
- **Third Equation of Motion:**
\[ v^2 = u^2 + 2as \]
Where:
- \( v \) is the final velocity,
- \( u \) is the initial velocity,
- \( a \) is the constant acceleration,
- \( s \) is the displacement.
These equations describe the relationships between position, velocity, acceleration, time, and displacement for an object moving in a straight line under constant acceleration. They are derived from the definitions of velocity and acceleration and are crucial for analyzing and predicting the motion of objects in many physical scenarios.