A particle starts from rest with an acceleration 2m/s. The acceleration of the particle decrees down to zero uniformly during the time interval of 4s. The velocity of the particle after 2s?

To determine the velocity of the particle after 2 seconds, we need to analyze the motion in two phases: the initial phase where the particle accelerates at a constant rate and the second phase where the acceleration decreases uniformly to zero. Let's break this down step by step.

Phase 1: Initial Acceleration

The particle starts from rest, meaning its initial velocity (u) is 0 m/s. It has a constant acceleration (a) of 2 m/s² for the first 4 seconds. To find the velocity after 2 seconds, we can use the formula:

v = u + at

• u = 0 m/s (initial velocity)
• a = 2 m/s² (acceleration)
• t = 2 s (time)

Plugging in the values:

v = 0 + (2 m/s²)(2 s) = 4 m/s

Phase 2: Deceleration to Zero

Now, we need to consider the next part of the motion where the acceleration decreases uniformly to zero over 4 seconds. Since we are only interested in the velocity after 2 seconds, we need to find out how the acceleration changes during this time.

The initial acceleration is 2 m/s², and it decreases to 0 m/s² over 4 seconds. This means the acceleration decreases at a rate of:

Rate of decrease = (Initial acceleration - Final acceleration) / Time

Rate of decrease = (2 m/s² - 0 m/s²) / 4 s = 0.5 m/s³

At 2 seconds, the acceleration will have decreased by:

Decrease in acceleration = Rate of decrease × Time = 0.5 m/s³ × 2 s = 1 m/s²

Thus, the acceleration at 2 seconds is:

Current acceleration = Initial acceleration - Decrease = 2 m/s² - 1 m/s² = 1 m/s²

Calculating the Velocity at 2 Seconds

Now that we have the acceleration at 2 seconds, we can find the velocity at that moment. The particle has already reached a velocity of 4 m/s after the first 2 seconds of constant acceleration. Now, we need to consider the effect of the new acceleration for the next 2 seconds (from 2 to 4 seconds).

Using the same formula for velocity:

v = u + at

• u = 4 m/s (velocity at 2 seconds)
• a = 1 m/s² (acceleration at 2 seconds)
• t = 2 s (time for the next phase)

Plugging in these values gives us:

v = 4 m/s + (1 m/s²)(2 s) = 4 m/s + 2 m/s = 6 m/s

Final Thoughts

After analyzing both phases of the particle's motion, we find that the velocity of the particle after 2 seconds is 4 m/s. However, if we consider the entire 4-second interval, the final velocity would be 6 m/s. This illustrates how the changing acceleration affects the overall motion of the particle.