A particle starts with an initial velocity of 10.0 ms$$^{-1}$$ along x-direction and accelerates uniformly at the rate of 2.0 m s$$^{-2}$$. The time taken by the particle to reach the velocity of 60.0 m s$$^{-1}$$ is ______.

A particle starts with initial velocity 10.0 m/s and accelerates uniformly at 2.0 m/s$$^2$$. We need to find the time to reach 60.0 m/s.

First, identify the known quantities.

Initial velocity: $$u = 10.0$$ m/s

Acceleration: $$a = 2.0$$ m/s$$^2$$

Final velocity: $$v = 60.0$$ m/s

Time: $$t = ?$$

Next, select the appropriate equation of motion.

Since we know $$u$$, $$v$$, and $$a$$, and need to find $$t$$, we use the first equation of motion:

$$ v = u + at $$

This equation relates velocity to time for uniformly accelerated motion. It follows directly from the definition of acceleration: $$a = \frac{v - u}{t}$$.

Now, substitute and solve for $$t$$.

$$ 60.0 = 10.0 + 2.0 \times t $$

$$ 2.0t = 60.0 - 10.0 = 50.0 $$

$$ t = \frac{50.0}{2.0} = 25\;\text{s} $$

The time taken is 25 s.

The correct answer is Option 1: 25 s.