A particle travels so that its displacement *D* (in metres) from its origin O is modelled by the equation, where *t* is the time in minutes after the particle has started to move.

a.       Determine the initial displacement.

b.       Determine the time when the velocity is first -20m/s

To solve this problem, let's first understand the given equation for displacement \( D(t) \), which represents the displacement of a particle as a function of time \( t \) (in minutes). However, the specific displacement function \( D(t) \) is not provided in your question. For generality, I'll assume the displacement is given by a polynomial function:

\[
D(t) = a t^n + b t^{n-1} + \cdots + c
\]

where \( a, b, c, \) etc., are constants, and \( t \) is time in minutes.

### a. Determine the Initial Displacement

The initial displacement refers to the displacement of the particle when \( t = 0 \). To find the initial displacement:

\[
D(0) = a \cdot 0^n + b \cdot 0^{n-1} + \cdots + c = c
\]

The initial displacement is simply the constant term \( c \) in the displacement function.

### b. Determine the Time When the Velocity is First -20 m/s

Velocity \( v(t) \) is the derivative of the displacement \( D(t) \) with respect to time \( t \):

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

To find the time when the velocity is first -20 m/s, we set the velocity equation equal to -20 and solve for \( t \):

\[
v(t) = -20
\]

However, since the exact form of \( D(t) \) is not given, please provide the specific displacement function \( D(t) \) for a more precise solution. 

Would you like to proceed with a generic example, or could you provide the displacement function?

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Here are five related questions:

1. What is the maximum displacement reached by the particle?
2. How does the acceleration of the particle change over time?
3. What is the total distance traveled by the particle after a certain period?
4. At what time is the particle momentarily at rest?
5. Can you determine the particle's displacement at any given time if its velocity function is known?

**Tip:** Knowing how to differentiate functions and solve equations is crucial for analyzing motion in physics.

Understand particle motion with a focus on initial displacement and velocity conditions. Learn how to determine the initial displacement and find the time when velocity first reaches -20 m/s. Calculus concepts and derivative formulas are used in this advanced high school level problem.

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