A car starts from rest and accelerates around a flat curve of radius \(R=36 \mathrm{~m}\). The tangential component of the car's acceleration remains constant at \(a_{\mathrm{t}}=3.3 \mathrm{~m} / \mathrm{s}^{2},\) while the centripetal acceleration increases to keep the car on the curve as long as possible. The coefficient of friction between the tires and the road is \(\mu=0.95 .\) What distance does the car travel around the curve before it begins to skid? (Be sure to include both the tangential and centripetal components of the acceleration.)

When an object moves in a circular path, it experiences a force that keeps it moving in that circle; this force is always directed towards the center of the circle, hence the term centripetal, which means 'center seeking'. This force causes an acceleration called centripetal acceleration.

Centripetal acceleration, denoted by the symbol \(a_c\), is given by the formula \(a_c = \frac{v^2}{R}\) where \(v\) is the velocity of the object, and \(R\) is the radius of the circular path.

In the context of the car problem, as the car speeds up, the velocity increases, and since the curve's radius \(R\) is constant, the centripetal acceleration must also increase in order to maintain the circular path. This explains why centripetal acceleration grows with increasing speed. When the car's centripetal acceleration surpasses the frictional force between the tires and the road, the car will begin to skid.

The coefficient of friction is a measure that describes how much frictional resistance is present between two surfaces in contact. It is a dimensionless quantity represented by the Greek letter \(\mu\). The value of \(\mu\) depends on the types of material in contact and can range from near zero (for very slippery surfaces) to greater than one (for very sticky surfaces).

In our car problem, the coefficient of friction between the tires and the road is 0.95, a relatively high number, indicating a good amount of grip. This coefficient is crucial as it determines the maximum centripetal force that can be sustained without skidding, which is directly tied to the frictional force \(F_f\) that can be calculated as \(F_f = \mu m_{car}g\), where \(m_{car}\) is the mass of the car and \(g\) is the acceleration due to gravity.

The implications of the coefficient of friction go beyond the maximum centripetal force; they also affect stopping distances, the ability to accelerate, and overall safety during motion.

While centripetal acceleration is directed towards the center of the circular path, tangential acceleration refers to the acceleration along the direction of motion - tangent to the curve. It represents how quickly the speed of the object is increasing or decreasing.

The formula for tangential acceleration, represented as \(a_t\), is simply the change in velocity over time along the tangential direction. For uniform acceleration, \(a_t = \frac{\Delta v}{\Delta t}\).

In the given car problem, the car starts from rest and accelerates around the curve with a constant tangential acceleration \(a_t = 3.3 \mathrm{m/s^2}\), which means the speed increases steadily. This tangential component determines how quickly the car can reach the critical speed at which the centripetal force equals the maximum frictional force available. The tangential acceleration is crucial for calculating the distance traveled before skidding occurs, using the kinematic equation \(s = \frac{1}{2}a_t t^2\), as seen in the solution.