A particle's position is r⃗ = (ct2-2dt3) i^ + (2ct2-dt3) j^, where c and d are positive constants. Find an expression for the time t > 0 when the particle is moving parallel to the x-axis.

t=4c/3d. For t>0

Step-by-step explanation:

Since we have to find an expression for the time t > 0 when the particle is moving parallel to x-axis, that implies there is no y component of velocity.

Hence,

Y component of velocity=0

We will find the y component of velocity via differentiating y-coordinate.

y = 2ct2-dt3

Differentiating y coordinate,

dy/dt = 4ct - 3dt²

Since y component of velocity is 0.

4ct - 3dt²=0

(4c - 3dt) t = 0

This is valid

when t=0 and when 4c-3dt=0.

4c=3dt

t=4c/3d

Since we are only asked for t>0, there is only one instant when the particle is moving int x-direction so the right answer is:

t = 4c/3d