As a 2.5 kg body moves in the positive direction along an `x` axis, a single force acts on it. The force is given by `F_(x)= -6x N`, with `x` in meters. The velocity at `x=3.5m` is 8.5 m/s. (a) Find the velocity of the body at `x=4.5` m. (b) Find the positive value of `x` at which the body has a velocity of 5.5 m/s.

Answer

To solve the problem, we will follow these steps:

### Given Data:
- Mass of the body, \( m = 2.5 \, \text{kg} \)
- Force acting on the body, \( F_x = -6x \, \text{N} \)
- Velocity at \( x = 3.5 \, \text{m} \), \( v = 8.5 \, \text{m/s} \)

### Part (a): Find the velocity of the body at \( x = 4.5 \, \text{m} \)

1. **Use Newton's Second Law**:
\[ F = ma \implies -6x = m \frac{dv}{dt} \]
Since \( a = \frac{dv}{dt} \) and we can express \( \frac{dv}{dt} \) in terms of \( \frac{dv}{dx} \):
\[ -6x = m v \frac{dv}{dx} \]

2. **Rearranging the equation**:
\[ v \, dv = -\frac{6}{2.5} x \, dx \]
\[ v \, dv = -2.4 x \, dx \]

3. **Integrate both sides**:
\[ \int v \, dv = \int -2.4 x \, dx \]
\[ \frac{v^2}{2} = -1.2 x^2 + C \]

4. **Find the constant \( C \)** using the initial condition \( v = 8.5 \, \text{m/s} \) at \( x = 3.5 \, \text{m} \):
\[ \frac{(8.5)^2}{2} = -1.2 (3.5)^2 + C \]
\[ 36.125 = -1.2 \cdot 12.25 + C \]
\[ 36.125 = -14.7 + C \implies C = 50.825 \]

5. **Substituting \( C \) back into the equation**:
\[ \frac{v^2}{2} = -1.2 x^2 + 50.825 \]
\[ v^2 = -2.4 x^2 + 101.65 \]

6. **Find \( v \) at \( x = 4.5 \, \text{m} \)**:
\[ v^2 = -2.4 (4.5)^2 + 101.65 \]
\[ v^2 = -2.4 \cdot 20.25 + 101.65 \]
\[ v^2 = -48.6 + 101.65 = 53.05 \]
\[ v = \sqrt{53.05} \approx 7.28 \, \text{m/s} \]

### Part (b): Find the positive value of \( x \) at which the body has a velocity of \( 5.5 \, \text{m/s} \)

1. **Set \( v = 5.5 \, \text{m/s} \)** in the equation:
\[ (5.5)^2 = -2.4 x^2 + 101.65 \]
\[ 30.25 = -2.4 x^2 + 101.65 \]
\[ -2.4 x^2 = 30.25 - 101.65 \]
\[ -2.4 x^2 = -71.4 \implies x^2 = \frac{71.4}{2.4} \]
\[ x^2 = 29.75 \implies x = \sqrt{29.75} \approx 5.45 \, \text{m} \]

### Final Answers:
- (a) The velocity of the body at \( x = 4.5 \, \text{m} \) is approximately \( 7.28 \, \text{m/s} \).
- (b) The positive value of \( x \) at which the body has a velocity of \( 5.5 \, \text{m/s} \) is approximately \( 5.45 \, \text{m} \).