A car is heading along a flat slippery road at a speed of 95 km/h. The minimum distance within which it can stop (because of ABS) is 66 m. Estimate the radius of the sharpest curve the car can negotiate on an icy flat surface at the same speed without skidding out.

Verified step by step guidance

1. Convert the speed of the car from km/h to m/s by using the conversion factor (1 km/h = 0.27778 m/s).
2. Use the formula for stopping distance to find the deceleration (a) using the equation $v^2=u^2+2as$, where $v$ is the final velocity (0 m/s), $u$ is the initial velocity (speed of the car in m/s), $s$ is the stopping distance, and $a$ is the acceleration (or deceleration in this case).
3. Apply the formula for centripetal force $F_c=\frac{m{v}^2}{r}$ where $m$ is the mass of the car, $v$ is the velocity in m/s, and $r$ is the radius of the curve. Since the force causing the centripetal acceleration is the frictional force, use $F_f=\mu mg$ where $\mu$ is the coefficient of friction and $g$ is the acceleration due to gravity.
4. Set the frictional force equal to the centripetal force to find the relationship between the coefficient of friction ($\mu$), the radius of the curve ($r$), and other known quantities. Solve for $r$ using the equation $r=\frac{v^2}{\mu g}$.
5. Estimate the coefficient of friction ($\mu$) for an icy road, which is typically low, and use it in the equation derived in the previous step to estimate the radius of the sharpest curve the car can negotiate without skidding.

Key Concepts

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, directed towards the center of the circle. For a car negotiating a curve, this force is provided by the friction between the tires and the road. If the required centripetal force exceeds the available frictional force, the car will skid out of the curve.

Friction and Skidding

Friction is the resistance that one surface or object encounters when moving over another. On a slippery surface, such as ice, the coefficient of friction is significantly lower, which reduces the maximum frictional force available to keep the car on its path. Understanding the relationship between speed, friction, and the potential for skidding is crucial for estimating safe turning radii.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. In this context, they can be used to relate the car's speed, the distance it can stop, and the radius of the curve. By applying these equations, one can calculate the necessary parameters to ensure the car can navigate the curve without losing traction.