Bonnie and Clyde are sliding a 325 kgkg bank safe across the floor to their getaway car. The safe slides with a constant speed if Clyde pushes from behind with 389 NN of force while Bonnie pulls forward on a rope with 325 NN of force.

**Part A**

What is the safe's coefficient of kinetic friction on the bank floor?

To find the coefficient of kinetic friction (\(\mu_k\)) between the safe and the bank floor, we can start by applying the principle of equilibrium. Since the safe is moving at a constant speed, the net force acting on it in the horizontal direction is zero. This implies that the total force exerted by Bonnie and Clyde is exactly balanced by the frictional force.

### Step 1: Identify the Forces
- **Clyde's pushing force:** \(F_{\text{Clyde}} = 389 \, \text{N}\) (acting forward)
- **Bonnie's pulling force:** \(F_{\text{Bonnie}} = 325 \, \text{N}\) (acting forward)
- **Total applied force:** \(F_{\text{applied}} = F_{\text{Clyde}} + F_{\text{Bonnie}} = 389 \, \text{N} + 325 \, \text{N} = 714 \, \text{N}\)
- **Frictional force (\(F_f\))** is the force opposing the motion.

### Step 2: Express the Frictional Force
The frictional force can be expressed as:
\[
F_f = \mu_k \cdot F_n
\]
where:
- \(F_n\) is the normal force, which equals the weight of the safe (\(mg\)) since the floor is horizontal.
- \(m\) is the mass of the safe (\(325 \, \text{kg}\)).
- \(g\) is the acceleration due to gravity (\(9.8 \, \text{m/s}^2\)).

So:
\[
F_n = mg = 325 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 3185 \, \text{N}
\]

### Step 3: Set Up the Equilibrium Equation
Since the safe moves with constant speed:
\[
F_{\text{applied}} = F_f
\]
\[
714 \, \text{N} = \mu_k \cdot 3185 \, \text{N}
\]

### Step 4: Solve for \(\mu_k\)
\[
\mu_k = \frac{714 \, \text{N}}{3185 \, \text{N}} \approx 0.224
\]

### Final Answer:
The coefficient of kinetic friction (\(\mu_k\)) between the safe and the bank floor is approximately **0.224**.

Would you like further explanations on any of these steps? Here are some related questions:

1. How would the coefficient change if Clyde applied more force?
2. What if the safe were on an inclined plane instead of a flat surface?
3. How does the mass of the safe affect the coefficient of kinetic friction?
4. What if the friction were static instead of kinetic?
5. How would the coefficient change if Bonnie applied less force?

**Tip:** Remember that the coefficient of kinetic friction is always less than or equal to the coefficient of static friction for the same two surfaces.

Learn how to calculate the coefficient of kinetic friction between a 325 kg bank safe and a bank floor. Understand the application of Newton's laws of motion and frictional force formulas. Suitable for high school students studying physics.

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