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