In Physics, Motion is the state of body in which it changes its position with time. Motion is fundamentally described by physical quantities such distance, displacement, speed, velocity, acceleration, and time. These physical quantities can be expressed in the form of a mathematical equation to express motion. These equations are called Equations of Motion. These equations can be derived via various methods such as Algebraic Method, Graphical Method and Calculus Method.

This article deals with the equation of motion and its derivation using the calculus method. This derivation is useful for class 11 students.

Fundamentals of Motion

The fundamentals of motion in physics are described by the concepts: of distance, displacement, speed, velocity, acceleration, and time. Here’s a brief overview:

Distance and Displacement

• Distance is the total length of the path travelled by an object, irrespective of direction. It is a scalar quantity.
• Displacement(s) is the change in position of an object, measured in a straight line from the starting point to the ending point. It is a vector quantity.

Speed and Velocity

• Speed is the rate at which an object covers distance. It is a scalar quantity and is calculated as the distance travelled divided by the time taken.
• Velocity(v) is the rate at which an object changes its position in a particular direction. It is a vector quantity and is calculated as the displacement divided by the time taken.

Acceleration

• Acceleration(a) is the rate at which velocity changes over time. Acceleration is also a vector quantity and is calculated as the change in velocity divided by the time taken.

Time

• Time(t) is a fundamental parameter used to measure the duration of an event or the interval between two events.

Equations of Motion by Calculus Method

There are three equations of motions also called Newton’s Equation of Motion. The expression of the three equations of motions are:

• First Equation of Motion: v = u + at
• Second Equation of Motion: s = ut + 1/2at²
• Third Equation of Motion: v² – u² = 2as

In the above three equations, v is final velocity, u is initial velocity, a is acceleration, s is distance and t is time.

Now, let’s derive these equation by calculus method:

First Equation of Motion by Calculus

Newton’s first equation of motion states that final velocity is equal to sum of initial velocity and product of acceleration and time. Mathematically, first equation is expressed as

v = u + at

Mathematically, acceleration is defined as

a = dv / dt

Integrating both sides of the equation with respect to time gives us:

^v∫_u dv= ^t∫₀ a dt

Where v is the final velocity, u is the initial velocity, a is the constant acceleration, and t is the time.

⇒ ^v∫_u dv= a^t∫₀ dt

⇒ v – u = at

Adding u on both sides, we get the final form

v = u + at

Second Equation of Motion by Calculus

Newton’s second equation of motion states that total distance covered is equal to sum of product of initial velocity and time and half of product of acceleration and square of time. Second equation of motion is mathematically expressed as s = ut + 1/2at²

Now, velocity is rate of change of distance. Hence, in terms of differentiation it can be expressed as

v = ds / dt

Integrating both sides of the equation with respect to time gives us:

^s∫₀ ds= ^t∫₀ v dt

Now using the first equation of motion

^s∫₀ ds= ^t∫₀ (u+ at) dt

Where v is the final velocity, u is the initial velocity, s is the displacement, a is the constant acceleration, and t is the time.

^s∫₀ ds = ^t∫₀ u dt + ^t∫₀ at dt

Since, u and a are constant

⇒ ^s∫₀ ds = u^t∫₀ dt + a^t∫₀ t dt

⇒ s = ut + at²/2

Hence, we derive the second equation of motion

s = ut+1/2 at²

Third Equation of Motion by Calculus

Third equation of motion states that difference between square of final velocity and initial velocity is equal to the twice the product of acceleration and distance. Mathematically, third equation of motion is given as: v² – u² = 2as

We start with the mathematical definition of acceleration

a = dv/dt

Multiply with v on both sides

av = v dv/dt

Substituting the definition of v = ds/dt

a ds/dt = v dv/dt

The time dependence can be removed here and this becomes

a ds = v dv

Integrating both sides

^s∫₀ a ds= ^v∫_u v dv

Where v is the final velocity, u is the initial velocity, s is the displacement, and a is the constant acceleration.

as = [v²/2]_u^v

(v² – u²)/2 = as

⇒ v² – u² = 2as

Rearranging, we get the Third Equation of Motion

v² = u²+2as

Applications of Calculus in Motion

Here are some applications of calculus in motion:

Analysing Uniformly Accelerated Motion

Calculus is used to derive the equations of motion for uniformly accelerated motion. It also enables us to determine instantaneous velocity and acceleration at any given point in time during the motion. By taking derivatives of displacement with respect to time, we obtain velocity functions. Similarly, taking derivatives of velocity with respect to time gives acceleration functions. Calculus allows us to understand how velocity and acceleration change over time. It helps in determining maximum velocity, maximum displacement, and time taken to reach specific positions.

Calculating Displacement and Velocity in Variable Accelerations

In cases of variable acceleration, calculus plays a crucial role in calculating displacement and velocity accurately. The velocity of the object at any time t is the integral of its acceleration function from the initial time t₀ to t:

v(t) = ^t∫_t0 a(t)dt

The displacement of the object from the initial position at time t is the integral of its velocity function from the initial time t₀ to t:

s(t) = ^t∫_t0 v(t)dt

Next step is to evaluate the integrals, and apply initial conditions to determine constants of integration. This would give the final form required.