A particle start movingrectilinear at t=0 such that it`s velocity as function of time is v equal to tsquare –t where v is in m per second and time is in seconds .the time interval in which the particle decelerates is

To determine the time interval during which the particle decelerates, we need to analyze the velocity function given by \( v(t) = t^2 - t \). Deceleration occurs when the velocity of the particle decreases over time, which means we need to find when the acceleration is negative. Acceleration is the derivative of velocity with respect to time.

Finding the Acceleration

First, let's calculate the acceleration \( a(t) \) by differentiating the velocity function:

• Given: \( v(t) = t^2 - t \)
• Acceleration: \( a(t) = \frac{dv}{dt} = \frac{d}{dt}(t^2 - t) \)
• Calculating the derivative: \( a(t) = 2t - 1 \)

Identifying Deceleration

Next, we need to find when the acceleration is negative:

• Set the acceleration less than zero: \( 2t - 1 < 0 \)
• Solving for \( t \):
• Rearranging gives \( 2t < 1 \)
• Dividing both sides by 2 results in \( t < \frac{1}{2} \)

Conclusion on the Time Interval

From our calculations, we find that the particle decelerates for the time interval \( t < \frac{1}{2} \) seconds. This means that during the first half of a second, the particle is slowing down. After this point, the acceleration becomes positive, indicating that the particle starts to accelerate again.

Visualizing the Motion

To visualize this, you can think of a car that starts moving forward. Initially, it might be pressing the brake pedal, causing it to slow down (decelerate). Once the driver releases the brake, the car begins to speed up again. In our case, the "brake" is the negative acceleration that occurs when \( t < \frac{1}{2} \) seconds.

In summary, the particle decelerates from \( t = 0 \) seconds up to \( t = \frac{1}{2} \) seconds, after which it begins to accelerate. This understanding of velocity and acceleration is crucial in analyzing motion in physics.