Question

A particle starts from origin with a velocity in (m/s) given by, dx/dt = x+1. The time in sec. taken by the particle to traverse a distance of 24m is what?

Class-12
Application of derivative​

Answer 1

ANSWER:

Given:

• Initial point = Origin
• Velocity is given by: dx/dt = x + 1

To Find:

• Time taken to traverse a distance of 24m.

Solution:

We are given that, velocity is given by,

$\implies \dfrac{dx}{dt}=x+1$

On rearranging,

$\implies \dfrac{dx}{x+1}=dt$

Now, we will integrate both sides.

$\displaystyle\implies \int\dfrac{dx}{x+1}=\int dt$

$\implies \log(x+1)=t + C$

As, the particle starts from 0, that means the value of x = 0 and t = 0.

$\implies \log(0+1)=0 + C$

$\implies \log(1)= C$

As, log (1) = 0,

$\implies C = 0$

Therefore, the path is independent of the value of C. So,

$\implies \log(x+1)= t$

Now, we need to find the time taken to traverse a distance of 24m.

So, value of x = 24.

$\implies \log(24+1)= t$

$\implies \log(25)= t$

$\implies \log(5^2)= t$

$\implies 2\log(5)= t$

Hence,

$\implies\bf t=2log(5)$

Therefore, time time taken to traverse a distance of 24m, by the particle is, 2log(5) seconds.