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?

Question

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

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Answer 1 · 2017-12-29T13:47:40

I get (a)

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

$vpropx$
$=>xpropv$