A particle starts from rest with acceleration $2\mathrm{m}/{\mathrm{s}}^2.$  The acceleration of the particle decreases down to zero uniformly during the time interval of 4 s. The velocity of the particle after 2 s is

In this problem, we are given that a particle starts moving with acceleration 2m/s² and that the acceleration decreases uniformly to zero over a period of 4 seconds. We are required to find the velocity of the particle after 2 seconds.

Step 1: Understanding the Variation of Acceleration

Since a particle starts moving with acceleration 2m/s² and the acceleration decreases uniformly to zero, we can describe the variation of acceleration as a linear function of time. This means that the acceleration is given by the equation:

a = mt + c

Where m is the slope of the acceleration-time graph, and c is the initial acceleration. Now, we can use the information provided in the problem to find the values of m and c.

Step 2: Finding the Value of c

At time t = 0, we know that the acceleration is 2 m/s². Therefore, by substituting these values into the equation for acceleration:

2 = m × 0 + c

From this equation, we get that c = 2.

Step 3: Finding the Value of m

Next, we are told that after 4 seconds, the acceleration has decreased to zero. At t = 4s, the acceleration is 0 m/s². Using this information, we substitute into the acceleration equation:

0 = m × 4 + 2

Solving for m, we get:

m = -1/2

Step 4: Equation for Acceleration

Now that we have both values for m and c, we can substitute them back into the original equation for acceleration:

a = -t/2 + 2

Step 5: Finding the Velocity after 2 Seconds

We know that velocity is the integral of acceleration with respect to time. Therefore, to find the velocity of the particle after 2 seconds, we can integrate the acceleration equation with respect to time:

∫(0 to v) dv = ∫(0 to 2) (-t/2 + 2) dt

Now, performing the integration:

v = 3 m/s

Final Answer

Thus, the velocity of the particle after 2 seconds is 3 m/s.

In conclusion, when a particle starts moving with acceleration 2m/s², and the acceleration decreases uniformly to zero in 4 seconds, the velocity of the particle after 2 seconds is 3 m/s.