Question

If a point moves in straight line in such a manner that its acceleration is proportional to its speed (a ∝ v) , what is the relation between distance covered and speed?

a) `x prop v`
b) `x prop v^2`
c) `x prop v^3`
d) `x prop sqrtv`

Answer 1

I get (a)

Explanation:

Given that acceleration `a prop v`, speed

From definition of acceleration and velocity
`=>(dv)/(dt)propdx/dt`
Integrating both sides with time `dt`, for LHS we get

`int(dv)/(dt)dt=v+c`
where `c` is a constant of integration and can be found from initial conditions.

For RHS we get

`intdx/dt dt=x+c_1`
where `c_1` is a constant of integration and can be found from initial conditions.

Assuming that initially at `t=0`, acceleration, velocity and displacement all are `=0`. Both `cand c_1` are`=0`. We get

`vpropx`
`=>xpropv`