Displacement x of a particle is a function of time x(t (sec)) = 2 + 6t + 2t2 meter. What is the velocity at time t = 5 sec?

To find the velocity of the particle at time \( t = 5 \) seconds, we need to first determine the velocity function, which is the derivative of the displacement function with respect to time.

Given the displacement function:
\[
x(t) = 2 + 6t + 2t^2
\]

We find the velocity \( v(t) \) by taking the derivative of \( x(t) \):

\[
v(t) = \frac{dx(t)}{dt}
\]

Let's calculate this derivative:

\[
v(t) = \frac{d}{dt} (2 + 6t + 2t^2) = 0 + 6 + 4t = 6 + 4t
\]

Now, substitute \( t = 5 \) seconds into the velocity function:

\[
v(5) = 6 + 4 \times 5 = 6 + 20 = 26 \, \text{m/s}
\]

So, the velocity of the particle at \( t = 5 \) seconds is \( 26 \, \text{m/s} \).

Would you like more details or have any questions?

Here are some related questions to consider:
1. How does the acceleration of the particle vary with time?
2. What is the initial velocity of the particle at \( t = 0 \)?
3. At what time will the velocity be zero, if ever?
4. How far does the particle travel in the first 5 seconds?
5. What is the displacement of the particle when the velocity is \( 10 \, \text{m/s} \)?

**Tip:** To find the acceleration, take the derivative of the velocity function.

Displacement x of a particle is a function of time x(t (sec)) = 2 + 6t + 2t^2 meter. What is the velocity at time t = 5 sec?

Learn how to find the velocity of a particle at t = 5 seconds given its displacement function x(t) = 2 + 6t + 2t^2. This solution uses calculus to take the derivative of the displacement function and apply it to solve for velocity.

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