Let's solve the problem step by step.

1. Determine the Initial Displacement

The initial displacement occurs at time $t = 0$.

Given the displacement equation: $D(t) = 24 + 15t - \frac{t^2}{2}$

To find the initial displacement, substitute $t = 0$ into the equation: $D(0) = 24 + 15(0) - \frac{0^2}{2} = 24$

So, the initial displacement is 24 meters.

2. Determine the Time When the Velocity is First 20 m/s

First, let's find the expression for velocity by differentiating the displacement equation $D(t)$ with respect to time $t$.

The velocity $V(t)$ is the derivative of $D(t)$: $V(t) = \frac{dD(t)}{dt}$

Differentiate the displacement function: $V(t) = \frac{d}{dt}\left(24 + 15t - \frac{t^2}{2}\right)$ $V(t) = 0 + 15 - t = 15 - t$

We need to find the time $t$ when the velocity $V(t)$ is 20 m/s: $20 = 15 - t$

Solve for $t$: $t = 15 - 20 = -5$

However, since time $t$ cannot be negative in this context, it means the particle never reaches a velocity of 20 m/s in the forward direction. The velocity reaches 20 m/s in the negative direction when $t = -5$, but this is not physically meaningful in the given context.

So, the particle does not reach a velocity of 20 m/s in the forward direction.

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Do you have any questions or need further details?

Here are some related questions you might find interesting:

1. What is the maximum velocity of the particle?
2. At what time does the particle come to rest?
3. What is the total distance traveled by the particle until it stops?
4. How long does it take for the particle to return to its initial position?
5. What is the acceleration of the particle at $t = 2$ minutes?

Tip: Remember that velocity is the derivative of displacement, and acceleration is the derivative of velocity, which provides insight into how the motion of the particle changes over time.