Application of Derivatives

Given:
Velocity v is proportional to the cube root of the displacement x.
v∝x1∕3
This means we can write:
v=kx1∕3
where k is a constant of proportionality.
We are asked to find how acceleration a depends on other quantities.
We know that:
a=‌

dv

dt

=‌

dv

dx

⋅‌

dx

dt

=v‌

dv

dx

Step 1: Find ‌

dv

dx

From v=kx1∕3,
‌

dv

dx

=k⋅‌

1

3

x−2∕3=‌

k

3

x−2∕3
Step 2: Substitute in acceleration formula:
a=v‌

dv

dx

=(kx1∕3)⋅(‌

k

3

x−2∕3)

Simplify:
a=‌

k2

3

x1∕3−2∕3=‌

k2

3

x−1∕3
Step 3: Express in terms of velocity if desired
Since v=kx1∕3⇒x1∕3=‌

v

k

,
x−1∕3=‌

k

v

So:
a=‌

k2

3

⋅‌

k

v

=‌

k3

3v

Therefore:
a∝‌

1

v

Final Answer:
Option B: inversely proportional to its velocity