Step by step integration of a=dv/dt to get v = u + at?

Question

Can someone please provide a step by step explanation of how you integrate

a= dv/dt

to get v = u + at



please can you explain exactly what your doing because i really dont follow this.


thanx =] x

Answer 1

First, you have to assume "a" is constant; otherwise it is not true that "v = u + at".

The equation "a = dv/dt" means that "a" is the derivative of "v" (with respect to time).

This means (by definition of derivative) that you can get "v" by taking the antiderivative (i.e. the indefinite integral) of "a".

To get the antiderivative of a constant, you just multiply the constant ("a") by the variable that you're differentiating with ("t"). Finally, you add some other arbitrary constant ("C") to the answer. (This is just elementary calculus.) So:

v = antiderivative(a)
= at + C

Now, what is the value of "C"? To find that out, set t=0:

When t=0
v = initial velocity = u
= a(0) + C
= C

In other words, C is just "u", the initial velocity. So you can substitute that into the equation to get:

v = u + at

Answer 2

dv/dt = a
dv = a dt
v = at + const



step by step:
multiply both sides of equation by dt:
dv = a dt

integrate both sides of equation:
integral of dv = a integral of dt
[integral of dx is always x plus constant]

finally:
v = at + const