Question

For the motion of particle , velocity v depends upon displacement x as v =20/(3x-2).If at t=0,x=0 then at what time t ,the x=20 ?

Answer 1

1. For the motion of particle, velocity $v$ depends upon displacement $x$ as $v = 20 / (3x - 2)$. If at $t = 0$, $x = 0$ then at what time $t$, the $x = 20$?

Let's integrate this equation by separating the variables.

$$\begin{array}{l} \frac{dx}{dt} = \frac{20}{3x - 2}, \\ (3x - 2)dx = 20dt, \\ \int_{0}^{2} (3x - 2)dx = \int_{0}^{t} 20dt, \\ \left(\frac{3x^2}{2} - 2x\right)\bigg|_{0}^{x} = 20t\big|_{0}^{x}, \\ \frac{3x^2}{2} - 2x = 20t. \end{array}$$

So, the time depends upon displacement as

$$t = \frac{1}{20} \left(\frac{3x^2}{2} - 2x\right).$$

Now, we can calculate the time, at which $x = 20$:

$$t_1 = \frac{1}{20} \left(\frac{3x_1^2}{2} - 2x_1\right).$$

Let evaluate the quantity.

$$t_1 = \frac{1}{20} \left(\frac{3 \cdot 20^2}{2} - 2 \cdot 20\right) = 28.$$

Answer: the time is equal to 28.