Equations of Motion

Motion is part of everyday life. A moving car, a falling ball, a running athlete, and a flying airplane are all examples of objects in motion. In physics, we use equations of motion to describe how an object moves when its acceleration remains constant.

Important Physical Quantities

Before studying the equations, we need to understand five important quantities.

Displacement, $(s)$
Displacement is the change in position of an object. It tells us how far the object has moved from its starting point in a particular direction.

Initial velocity, $(u)$
Initial velocity is the velocity of an object at the beginning of its motion.

Final velocity, $(v)$
Final velocity is the velocity of an object after a certain time.

Acceleration, $(a)$
Acceleration is the rate at which velocity changes with time. If an object speeds up, slows down, or changes direction, it is accelerating.

Time, $(t)$
Time is the duration for which the object is moving. These equations are valid only when the acceleration is constant.

The Three Equations of Motion

For motion in a straight line with constant acceleration, the three main equations are;

Here,
$u$ = initial velocity
$v$ = final velocity
$a$ = acceleration
$t$ = time
$s$ = displacement

Derivation of the First Equation of Motion

Acceleration is defined as the change in velocity divided by time;

Multiplying both sides by $t$, we get;

Now rearrange the equation: \[\boxed{v=u+at}\tag{1.4}\]

This is the first equation of motion. The first equation tells us the final velocity of an object after a given time.

Derivation of the Second Equation of Motion

For constant acceleration, the average velocity is;

Displacement is equal to average velocity multiplied by time:

Substituting the value of average velocity:

Using the first equation; $v=u+at$
Substitute this value of $v$:

Derivation of the Third Equation of Motion

Start with the first equation:$$v = u + at$$

From this equation,$$t = \frac{v-u}{a}$$

We also know that displacement is:$$s = \frac{u+v}{2}t$$

Substitute the value of $t$:$$s = \frac{u+v}{2} \times \frac{v-u}{a}$$$$s = \frac{(v+u)(v-u)}{2a}$$

Using the algebraic identity,$$(v+u)(v-u) = v^2-u^2$$

we get:$$s = \frac{v^2-u^2}{2a}$$

Multiplying both sides by $2a$:$$2as = v^2-u^2$$

Rearranging:$$v^2 = u^2 + 2as$$

This is the third equation of motion. The third equation is useful when time is not given in the problem.

Simple Example

A car starts from rest and accelerates at $2 \, \text{m/s}^2$ for $5 \, \text{s}$. Find its final velocity.

Given:$$u = 0$$

Using the first equation:$$v = u + at$$

Substitute the values:$$v = 0 + (2)(5)$$

Therefore, the final velocity of the car is:$10 m/s.$

The car reaches a final velocity of 10 m/s after 5 seconds.

Why Equations of Motion Are Important

Equations of motion help us solve many real-life problems. They are used in road safety, sports, vehicle design, space science, and engineering. For example, these equations can help calculate how long a car takes to stop, how far a ball travels, or how fast an object is moving before it reaches the ground. <div class=”statement-box”> In short, equations of motion help us predict the position and velocity of a moving object. </div>

References

1. NCERT Science Textbook, Class 9, Chapter: Motion.
2. OpenStax, College Physics, Chapter: Kinematics.
3. The Physics Classroom, One-Dimensional Kinematics.