Question

How to prove v = u + at?

Answer 1

v = u+ at :

Good question,
Here is your perfect answer!

Since a = (v-u) /t

=) at = (v-u)

=) at + u = v

=) v = u + at.

Answer 2

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$\textbf {FIRST \: EQUATION \: OF \: MOTION }$


a=(v-u)/t

at = v-u

=> v-u = at

=> v= u + at (1)

This is Newton's First equation of motion. As you can you see, we can use this equation to calculate the velocity of a body which underwent an acceleration of a m/s for a time period of t seconds, provided we know the initial velocity of the body. Initial velocity i.e. u is the velocity of the body just before the body started to accelerate i.e. the velocity at t=0.

SECOND EQUATION OF MOTION

velocity = distance traveled / time taken

average velocity = (u+v)/2

.: (u+v)/2 = s/t

s = [(u+v)/2]t

From equation (1) we have v=u+at, substituting this in the above equation for v, we get

s = [(u+u+at)/2]t

=> s = [(2u+at)/2]t

=> s = [(u + (1/2)at)]t

=> s = ut + (1/2)at2 - (2)

This is Newton's second equation of Motion. This equation can be used to calculate the distance traveled by a body moving with a uniform acceleration in a time t. Again here, if the body started from rest, then we shall substitute u=0 in this equation

THIRD EQUATION OF MOTION

We start with squaring equation (1). Thus we have

v² = (u+at)²

=> v² = u² + a2t2 + 2uat

=> v² = u² + 2uat + a2t2

=> v² = u2 + 2a(ut + (1/2)at2)

now, using equation 2 we have

=> v² = u² + 2as - (3)

As you can see, the above equation gives a relation between the final velocity v of the body and the distance s traveled by the body.

Thus, we have the the three Newton's equations of Motion as

 

1) v= u + at

 

2) s = ut + (1/2)at²