Finding the formula for displacement of a moving body by integration.
By differentiating s = ut + (at^2)/2 w. r. to t, we obtain
ds/dt = u + 2(at)/2
v= u + at, where ds/dt = v - velocity of a moving body.
Also, dv/dt = a, where dv/dt denotes the rate of change of velocity with respect to time and a denote acceleration of a moving body.
Derivation of Kinetic Energy(K.E) from v =u + at.
Squaring: v^2 = (u+at)^2 = u^2 + 2uat + (at)^2 = u^2 + 2a{ut +a (t^2)2}.
i.e. v^2 = u^2 + 2as, where s = ut +a (t^2)2.
Now, a =( v^2 - u^2)/2s ⇒ ma= m( v^2 - u^2)/2s ⇒ F = m( v^2 - u^2)/2s, where F is force.
Next, work done on the body is given by
W=F *s = s * m( v^2 - u^2)/2s = (mv^2)/2 - (mu^2)/2, where W = Force * Displacement.
Put the initial velocity u = 0. Then, K.E = (mv^2)/2 .
If K is the kinetic energy, dK is the kinetic energy per second, which is equal to mvdv, that is, dK= mvdv. By integrating this equation on both sides, we get
K = (mv^2)/2.
- Badges
- Science topic