If the only forces acting on a 3.0-kg mass are F1 = (3i - 8j) N and F2 = (5i + 3j) N, what is the magnitude of the acceleration (in m/s2) of the particle?

To find the magnitude of the acceleration, we’ll use Newton’s second law: 

\[
\vec{F}_{\text{net}} = m \vec{a}
\]

where \(\vec{F}_{\text{net}}\) is the net force acting on the object, \(m\) is the mass, and \(\vec{a}\) is the acceleration. 

### Step 1: Find the Net Force
Given forces:
- \( \vec{F}_1 = (3i - 8j) \, \text{N} \)
- \( \vec{F}_2 = (5i + 3j) \, \text{N} \)

The net force \(\vec{F}_{\text{net}}\) is the sum of \(\vec{F}_1\) and \(\vec{F}_2\):
\[
\vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2 = (3 + 5)i + (-8 + 3)j
\]
\[
\vec{F}_{\text{net}} = 8i - 5j \, \text{N}
\]

### Step 2: Calculate the Magnitude of the Net Force
The magnitude of \(\vec{F}_{\text{net}}\) is:
\[
|\vec{F}_{\text{net}}| = \sqrt{(8)^2 + (-5)^2}
\]
\[
|\vec{F}_{\text{net}}| = \sqrt{64 + 25} = \sqrt{89} \approx 9.43 \, \text{N}
\]

### Step 3: Calculate the Acceleration
Using Newton's second law, \( |\vec{F}_{\text{net}}| = m|\vec{a}|\), we can solve for the magnitude of acceleration \( |\vec{a}| \):
\[
|\vec{a}| = \frac{|\vec{F}_{\text{net}}|}{m} = \frac{9.43}{3.0}
\]
\[
|\vec{a}| \approx 3.14 \, \text{m/s}^2
\]

### Final Answer
The magnitude of the acceleration is approximately:
\[
\boxed{3.14 \, \text{m/s}^2}
\]

Would you like a further breakdown of these steps or have any questions?

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**Related Questions:**

1. How would the acceleration change if the mass was doubled?
2. What would the net force be if \(\vec{F}_2\) were reversed?
3. How do you find acceleration if only given the magnitude of forces?
4. What would be the direction of the acceleration vector in terms of angles?
5. How would adding a third force impact the solution?

**Tip:** When solving vector problems, breaking them down into components often simplifies the calculations.

Find the magnitude of acceleration for a 3.0-kg mass with forces F1 = (3i - 8j) N and F2 = (5i + 3j) N. This solution demonstrates Newton's Second Law, vector addition, and vector magnitude calculations, ideal for Grades 10-12 physics.

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