Question

Calculus method for deriving 2 equations of motion

Answer 1

Newton's second equation of motion :-

S=ut+12at2 [where, u is the initial velocity, a is the acceleration and t is the time interval]

This Equation simply finds a relation between distance travelled by a particle (classically) under uniform acceleration.

Follow the statement I said in italics. In order to find the distance covered by the particle, we need to find some equations to start with. Right??

So let's see what pieces of information (bundles of equations) do we have with us, initially.

We have, a very primary equation with us, dSdt=v…(I)

(Considering motion in a straight line only)

And we also have the equation

dvdt=a…(II)

Simply replacing the v in eqn (II) by eqn (I), we find

d2Sdt2=a…(III)

This is what we need to solve. It's easy.

You know, d2Sdt2=ddt(dSdt)=a

⟹dSdt=∫adt=at+c1

Since, dSdt is the velocity of the particle,

Therefore, at t = 0, dSdt|t=0=u

⟹u=a∗0+c1=c1

⟹c1=u

Therefore, dSdt=u+at

Thus, S=∫(udt+atdt)

⟹S=ut+12at2+c2

If say, the particle is already having a displacement S0 the moment you start measuring it's motion. Then, at t = 0, S=S0

This makes S=S0+ut+12at2

Since, in most of the practical cases, we start measuring a motion when the particle starts displacing (i.e., when S0=0),

We get

S=ut+12at2

Answer 2

here's your answer dear