the velocity of a moving particle along a straight line is given by v=1 over 288 t[576-t squared] find the distance travelled in the 24 seconds

To find the distance traveled by the particle over the 24-second interval, we can use the velocity function \( v(t) \) and integrate it with respect to time to get the displacement.

The velocity function is given as:

\[
v(t) = \frac{1}{288} t (576 - t^2)
\]

To find the distance traveled from \( t = 0 \) to \( t = 24 \), we integrate \( v(t) \) over this interval:

\[
s = \int_{0}^{24} v(t) \, dt = \int_{0}^{24} \frac{1}{288} t (576 - t^2) \, dt
\]

Let's compute the integral step by step.

### 1. Expand the expression inside the integral:

\[
v(t) = \frac{1}{288} t (576 - t^2) = \frac{1}{288} (576t - t^3)
\]

Now the integral becomes:

\[
s = \frac{1}{288} \int_{0}^{24} (576t - t^3) \, dt
\]

### 2. Split the integral:

\[
s = \frac{1}{288} \left[ \int_{0}^{24} 576t \, dt - \int_{0}^{24} t^3 \, dt \right]
\]

### 3. Integrate each term:

- The integral of \( 576t \) is:

\[
\int 576t \, dt = 288t^2
\]

Evaluating this from 0 to 24 gives:

\[
288(24^2) - 288(0^2) = 288(576) = 165888
\]

- The integral of \( t^3 \) is:

\[
\int t^3 \, dt = \frac{t^4}{4}
\]

Evaluating this from 0 to 24 gives:

\[
\frac{24^4}{4} - \frac{0^4}{4} = \frac{331776}{4} = 82944
\]

### 4. Combine the results:

Now, subtract the integrals and multiply by \( \frac{1}{288} \):

\[
s = \frac{1}{288} \left( 165888 - 82944 \right) = \frac{1}{288} (82944) = 288
\]

Thus, the distance traveled by the particle in 24 seconds is **288 units**.

---

Would you like any further clarifications or additional details? Here are five related questions:

1. What is the difference between displacement and distance traveled?
2. How does integrating velocity give the displacement?
3. How would the result change if the velocity function had a different form?
4. Can this technique be applied to a velocity function with trigonometric terms?
5. How do we interpret negative velocity in terms of displacement?

**Tip:** When velocity changes direction, be mindful of whether you're asked for total distance (which is always positive) or displacement (which can be negative).

The velocity of a moving particle along a straight line is given by v = (1/288) * t * [576 - t^2]. Find the distance traveled in 24 seconds.

Learn how to calculate the distance traveled by a particle in 24 seconds using a given velocity function: v = (1/288) * t * (576 - t^2). This problem involves calculus concepts such as definite integrals and kinematics. Suitable for high school and college-level students.

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