Force and Motion

Distance and Displacement

The length between two points is called distance, and its SI unit is the meter (m). Distance is a scalar quantity.

The shortest straight-line distance between two points is called displacement. Its SI unit is also the meter (m), but it is a vector quantity because it has both magnitude and direction.

Speed and Velocity

The distance traveled by a body in unit time is called speed. Its SI unit is the meter per second or m/s. Speed is a scalar quantity.

The distance traveled by a body in unit time in a specific direction is called speed. Its SI unit is the meter per second or m/s. Speed is a scalar quantity.

Acceleration

The rate of increase in velocity is called acceleration. It is represented by the letter ‘a’ and its SI unit is meter per second or m/s².

\(\therefore a = \frac{(v-u)}{2}\) and \(\therefore a = \frac{(v+u)}{2}\)

The rate of decrease in velocity is called retardation. It is represented by the letter negative of acceleration or ‘-a’.

Equations of Linear Motion

Before we derive these equations, let's define the key terms:

- u: Initial velocity (the velocity at the start of the motion)

- v: Final velocity (the velocity after time \( t \))

- a: Constant acceleration

- t: Time taken

- s: Displacement (the distance covered during the motion)

1. First Equation of Motion

The first equation is: \(v = u + at\)

This equation relates an object’s final velocity to its initial velocity, acceleration, and time. Here’s how we derive it:

Derivation:

Acceleration is defined as the rate of change of velocity. Mathematically:

\(a = \frac{v - u}{t}\)

Rearranging this equation to solve for the final velocity \( v \):

\(\therefore v = u + at\)

This is the first equation of motion, which gives the object's velocity after time \( t \) when it is moving with constant acceleration.

2. Second Equation of Motion

The Second Equation is: \(s = ut + \frac{1}{2}at^2\)

This equation describes the displacement of an object after a certain time under constant acceleration.

Derivation:

The displacement \( s \) is the distance covered by an object. For motion with constant acceleration, we calculate displacement using the average velocity:

\(\text{Average velocity} = \frac{u + v}{2}\)

Displacement \( s \) is given by:

\(s = \text{Average velocity} \times t = \left( \frac{u + v}{2} \right) t\)

Substituting \( v = u + at \) from the first equation into this equation:

\(s = \left( \frac{u + (u + at)}{2} \right) t\)

Simplifying:

\(\text{or,} s = \left( \frac{2u + at}{2} \right) t \)

\(\therefore s = ut + \frac{1}{2}at^2\)

This is the second equation of motion, which gives the object’s displacement after time \( t \).

3. Third Equation of Motion

The third equation is: \(v^2 = u^2 + 2as\)

The third equation relates the final velocity to the initial velocity, acceleration, and displacement.

Derivation:

Starting with the first equation:

\(v = u + at\)

Solve for time \( t \):

\(t = \frac{v - u}{a}\)

Substitute this value of \( t \) into the second equation:

\(s = ut + \frac{1}{2}at^2\)

Now, we substitute \( t = \frac{v - u}{a} \) into this equation:

\(s = u \frac{v - u}{a} + \frac{1}{2} a \frac{(v - u)^2}{a^2}\)

Simplifying:

\(s = \frac{u(v - u)}{a} + \frac{(v - u)^2}{2a}\)

Multiply both sides by \( 2a \) to eliminate the denominators:

\(\text{or,} 2as = 2u(v - u) + (v - u)^2\)

\(\text{or,} 2as = v^2 - u^2\)

\(\therefore v^2 = u^2 + 2as \)

This is the third equation of motion, which relates the final velocity \( v \) to the initial velocity \( u \), displacement \( s \), and acceleration \( a \).