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, we need to find the velocity function by differentiating the displacement function $D(t)$ with respect to time $t$.

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

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

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

Solve for $t$: $t = 15 + 20 = 35$

So, the time when the velocity first reaches $-20$ m/s is 35 minutes.

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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 time when the particle comes to rest?
2. What is the maximum displacement of the particle from the origin?
3. At what time does the particle return to the initial displacement of 24 meters?
4. What is the acceleration of the particle at $t = 10$ minutes?
5. What is the total distance traveled by the particle until it reaches a velocity of $-20$ m/s?

Tip: In kinematics, understanding the relationship between displacement, velocity, and acceleration is crucial for analyzing the motion of particles. The velocity function gives you information about how fast and in what direction the particle is moving at any given time.