The position of a golf ball moving uphill along a gently sloped green is given as a function time t by : `x=2+8t-3t^(2)`

Options

A. At t=0, `vecF_(1)=50hat(i),vecf_(2)`=0
B. At t=2 sec ,`vecf_(1)= 50 hat(i), vecf_(2)=-10hat(i)`
C. At t= 4sec. `vecf_(1)=30hat(i),vecf_(2)=-20hat(i)`
D. At t= 5sec both blocks start movingt in + ive x-direction

Answer

To solve the problem regarding the position of a golf ball moving uphill along a gently sloped green, we start with the given function:

\[ x(t) = 2 + 8t - 3t^2 \]

This function describes the position \( x \) of the golf ball at any time \( t \).

### Step 1: Find the maximum position of the golf ball
To find the maximum position of the golf ball, we need to determine the vertex of the quadratic function. The general form of a quadratic function is:

\[ ax^2 + bx + c \]

In our case, we have:

- \( a = -3 \)
- \( b = 8 \)
- \( c = 2 \)

The time \( t \) at which the maximum position occurs can be found using the formula:

\[ t = -\frac{b}{2a} \]

### Step 2: Calculate \( t \)
Substituting the values of \( a \) and \( b \):

\[ t = -\frac{8}{2 \times -3} = \frac{8}{6} = \frac{4}{3} \]

### Step 3: Find the maximum position \( x \)
Now, we substitute \( t = \frac{4}{3} \) back into the position function to find the maximum position \( x \):

\[ x\left(\frac{4}{3}\right) = 2 + 8\left(\frac{4}{3}\right) - 3\left(\frac{4}{3}\right)^2 \]

Calculating each term:

1. \( 8 \left(\frac{4}{3}\right) = \frac{32}{3} \)
2. \( \left(\frac{4}{3}\right)^2 = \frac{16}{9} \) and \( 3 \left(\frac{4}{3}\right)^2 = 3 \times \frac{16}{9} = \frac{48}{9} = \frac{16}{3} \)

Now substituting these back into the equation:

\[ x\left(\frac{4}{3}\right) = 2 + \frac{32}{3} - \frac{16}{3} \]

Combine the terms:

\[ x\left(\frac{4}{3}\right) = 2 + \frac{32 - 16}{3} = 2 + \frac{16}{3} \]

Convert 2 to a fraction with a denominator of 3:

\[ 2 = \frac{6}{3} \]

Now add:

\[ x\left(\frac{4}{3}\right) = \frac{6}{3} + \frac{16}{3} = \frac{22}{3} \]

### Final Answer
The maximum position of the golf ball is:

\[ \boxed{\frac{22}{3}} \]