Let's break down and solve the problem step by step. --- ### **Given:** - **Radius of curve, \( r = 70 \) m** - **Coefficient of friction, \( \mu = 0.4 \)** - **Acceleration due to gravity, \( g = 9.8 \) m/s²** --- ### **Step 1: Find the maximum frictional force** The frictional force provides the centripetal force required to keep the car moving in a circle: \[ F_{\text{friction}} = \mu F_N = \mu mg \] --- ### **Step 2: Set frictional force equal to centripetal force** Centripetal force needed: \[ F_c = \frac{mv^2}{r} \] Set \( F_c = F_{\text{friction}} \): \[ \frac{mv^2}{r} = \mu mg \] --- ### **Step 3: Solve for the maximum speed \( v \)** \[ \frac{mv^2}{r} = \mu mg \] Divide both sides by \( m \): \[ \frac{v^2}{r} = \mu g \] Multiply both sides by \( r \): \[ v^2 = \mu g r \] Take the square root: \[ v = \sqrt{\mu g r} \] --- ### **Step 4: Substitute the values** \[ v = \sqrt{0.4 \times 9.8 \times 70} \] Calculate inside the square root: \[ 0.4 \times 9.8 = 3.92 \] \[ 3.92 \times 70 = 274.4 \] So, \[ v = \sqrt{274.4} \approx 16.56\ \text{m/s} \] --- ## **Final Answer:** \[ \boxed{v_{\text{max}} \approx 16.6\ \text{m/s}} \] --- **Explanation:** The maximum speed is determined by the maximum frictional force, which provides the necessary centripetal force to round the curve without sliding. If the speed exceeds this value, the car will lose traction and possibly skid.

The maximum speed of a vehicle on a curved path is limited by the frictional force, which provides the necessary centripetal force to maintain circular motion. When the vehicle's speed exceeds this limit, friction cannot prevent slipping, causing the vehicle to slide outward.