Question

derivation of second and third law of motion by calculus method​

Answer 1

bro please, refer from your text book

Answer 2

Answer:

Explanation:

We know that the acceleration of the body is defined as the rate of change of velocity.

Mathematically, acceleration is represented as follows:

a=v−ut

where v is the final velocity and u is the initial velocity.

Rearranging the above equation, we arrive at the first equation of motion as follows:

v=u+at

The first equation of motion can be derived using a velocity-time graph for a moving object with an initial velocity of u, final velocity v, and acceleration a.The velocity of the body changes from A to B in time t at a uniform rate.

BC is the final velocity and OC is the total time t.

A perpendicular is drawn from B to OC, a parallel line is drawn from A to D, and another perpendicular is drawn from B to OE (represented by dotted lines).

Following details are obtained from the graph above:

The initial velocity of the body, u = OA

The final velocity of the body, v = BC

From the graph, we know that

BC = BD + DC

Therefore, v = BD + DC

v = BD + OA (since DC = OA)

Finally,

v = BD + u (since OA = u) (Equation 1)

Now, since the slope of a velocity-time graph is equal to acceleration a,

So,

a = slope of line AB

a = BD/AD

Since AD = AC = t, the above equation becomes:

BD = at (Equation 2)

Now, combining Equation 1 & 2, the following is obtained:

v = u + at

Derivation of First Equation of Motion by Calculus Method

Since acceleration is the rate of change of velocity, it can be mathematically written as:

a=dvdt

Rearranging the above equation, we get

adt=dv

Integrating both the sides, we get

∫t0adt=∫vudv

at=v−u

Rearranging, we get

v=u+at

Derivation of Second Equation of Motion

For the derivation of the second equation of motion, consider the same variables that were used for derivation of the first equation of motion.

Derivation of Second Equation of Motion by Algebraic Method

Velocity is defined as the rate of change of displacement. This is mathematically represented as:

Velocity=DisplacementTime

Rearranging, we get

Displacement=Velcoity×Time

If the velocity is not constant then in the above equation we can use average velocity in the place of velocity and rewrite the equation as follows:

Displacement=(InitialVelocity+FinalVelocity2)×Time

Substituting the above equations with the notations used in the derivation of the first equation of motion, we get

s=u+v2×t

From the first equation of motion, we know that v = u + at. Putting this value of v in the above equation, we get

s=u+(u+at))2×t

s=2u+at2×t

s=(2u2+at2)×t

s=(u+12at)×t

On further simplification, the equation becomes:

s=ut+12at2