Saved Bookmarks
| 1. |
Derive equations of uniformly acceleration motion by calculus method. |
|
Answer» Solution :Consider an object moving in a straight line with uniform or constant acceleration 'a'. Let u be the velocity of the object at time t=0, and v be velocity of the body at a later time t. Velocity - time relation (i) The acceleration of the body at any instant is given by the first derivative of the velocity with respect to time, `a= (dv)/(dt)" or "dv=a.dt` Integrating both sides with the CONDITION that as time changes from 0 to t, the velocity changes from u to v. For the constant acceleration, `int_(u)^(v) dv= int_(0)^(t) a dt = a int_(u)^(v) dt implies [v]_(u)^(v) = a [t]_(0)^(t)"""............."(1)` `v-u = at" or " v= u+at` DISPLACEMENT - time relation (ii) The velocity of the body is given by the first derivative of the displacement with respect to time. `v= (ds)/(dt)" or "ds = vdt` and since `v=u+at` We get `ds = (u+at)dt` Assume that initially at time t=0, the particle started from the origin. At a later time t, the particle displacement is s. Further assuming that acceleration is time - independent, we have `int_(0)^(s) ds= int_(0)^(t) u dt +int_(0)^(t) at dt " or "s =ut +(1)/(2) at^2"""............."(2)` Velocity -displacement relation (iii) The ACCELARATION is given by the first derivative of velocity with respect to time. `a= (dv)/(dt)=(dv)/(ds)(ds)/(dt)= (dv)/(ds)v` [ since `(ds)/(dt)=v`] where s is distance traversed] This is rewritten as `a=(1)/(2) (dv^2)/(ds)" or " ds = (1)/(2a) d(v^2)` Integrating the above equation, using the FACT when the velocity changes from `u^2" to " v^2`, displacement changes from 0 to s, we get `int_(0)^(s) ds = int_(u)^(v) (1)/(2a)d (v^2)` `:. S= (1)/(2a) (v^2-u^2)` `:. v^2=u^2+2as"""............."(3)` We can also derive the displacement 's' in TERMS of initial velocity 'u' and final velocity v. From equation 1, we can write `at = v-u` Substitute this in equation 2, we get `S= ut +(1)/(2)(v-u)t` `S= ((u+v)t)/(2)""".............."(4)` The equations 1,2,3 and 4 are called kinematic equations of motion, and have a wide variety of practical applications. Kinematic equations `v= u+at` `S= ut +(1)/(2) at^2` `v^2= u^2+2as` `s=((u+v)t)/(2)` It is to be noted that all these kinematic equations are valid only if the motion is in a straight line with cosntant acceleration. For circular motion and oscillatory motion these equations are not applicable. |
|