1. Why Do We Need Equations of Motion?
When an object moves with uniform acceleration, its speed changes at a steady rate. To describe such motion, we use three special formulas called the equations of motion.
These equations help us find distance, time, acceleration, initial velocity, or final velocity without needing complicated calculations.
1.1. Uniform Acceleration
Uniform acceleration means the velocity changes by the same amount every second. For example, a car increasing its speed steadily by 2 m/s every second.
1.2. Variables Used in Equations of Motion
- \( u \): initial velocity
- \( v \): final velocity
- \( a \): acceleration
- \( t \): time taken
- \( s \): displacement
2. First Equation of Motion
The first equation gives the relationship between velocity, initial velocity, acceleration, and time.
Equation:
\( v = u + at \)
2.1. Meaning of the First Equation
This equation shows that final velocity increases by \( at \) when the object has constant acceleration.
2.2. Example
A bike starts at 4 m/s and accelerates at 2 m/s² for 3 seconds:
\( v = 4 + (2 \times 3) = 10 \, \text{m/s} \)
3. Second Equation of Motion
The second equation helps us find displacement when an object is accelerating uniformly.
Equation:
\( s = ut + \dfrac{1}{2}at^2 \)
3.1. Meaning of the Second Equation
This equation adds two effects: distance covered due to initial velocity and distance added due to acceleration.
3.2. Example
If a car starts at rest (\( u = 0 \)) and accelerates at 3 m/s² for 4 seconds:
\( s = 0 \cdot 4 + \dfrac{1}{2} (3)(4^2) = 24 \, \text{m} \)
4. Third Equation of Motion
The third equation connects velocity and displacement without involving time.
Equation:
\( v^2 = u^2 + 2as \)
4.1. Meaning of the Third Equation
This equation is useful when time is not known or cannot be measured easily.
4.2. Example
A ball is thrown upwards with a velocity of 20 m/s. At the highest point, its velocity becomes 0:
\( 0 = 20^2 + 2(-9.8)s \)
\( s = 20.4 \, \text{m (approximately)} \)
5. When Can We Use the Equations of Motion?
The equations of motion can be used only when acceleration is uniform, meaning it does not change over time.
5.1. Valid Situations
- An object moving under constant gravity (ignoring air resistance).
- A vehicle accelerating steadily on a straight road.
- A ball rolling down a smooth slope.
5.2. Situations Where They Cannot Be Used
- Motion on rough or uneven surfaces where acceleration changes.
- Objects moving with random or irregular speeds.
- Motion involving sharp turns or changing directions rapidly.
6. Using Equations of Motion in Real Life
These equations help us solve many practical problems such as predicting stopping distances of vehicles, calculating how long a rocket takes to reach a certain speed, or estimating how high a ball will rise when thrown upward.
6.1. Simple Real-Life Example
If a car moving at 15 m/s applies brakes and slows down with an acceleration of \( -2 \, \text{m/s}^2 \), we can find how long it takes to stop using:
\( v = u + at \)
\( 0 = 15 + (-2)t \)
\( t = 7.5 \, \text{s} \)