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 \( a \).
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