Problem 90 - Algebra and Trigonometry

Question

Speed of a Skidding Car Police use the formula \(s=\sqrt{30 f d}\) to estimate the speed \(s(\text { in } \mathrm{mi} / \mathrm{h})\) at which a car is traveling if it skids \(d\) feet after the brakes are applied suddenly. The number \(f\) is the coefficient of friction of the road, which is a measure of the "slipperiness" of the road. The following table gives some typical estimates for \(f\) $$ \begin{array}{|c|c|c|c|}\hline & {\text { Tar }} & {\text { Concrete }} & {\text { Gravel }} \\ \hline \text { Dry } & {1.0} & {0.8} & {0.2} \\ {\text { Wet }} & {0.5} & {0.4} & {0.1} \\ \hline\end{array} $$ (a) If a car skids 65 \(\mathrm{ft}\) on wet concrete, how fast was it moving when the brakes were applied? (b) If a car is traveling at 50 \(\mathrm{mi} / \mathrm{h}\) , how far will it skid on wet tar?

Step-by-Step Solution

Answer

(a) 27.93 mi/h; (b) 166.67 ft.

Step 1: Identify the Data for Part (a)

Identify the values given in part (a) of the problem. The problem states that the car skids 65 feet on wet concrete. From the table, the coefficient of friction \( f \) for wet concrete is 0.4.

Step 2: Use the Speed Formula for Part (a)

Use the formula \( s = \sqrt{30 f d} \) to calculate the speed. Substitute \( f = 0.4 \) and \( d = 65 \) into the formula.

Step 3: Calculate Speed for Part (a)

Substitute the values into the equation: \[ s = \sqrt{30 \times 0.4 \times 65} \] Calculate the expression inside the square root: \( 30 \times 0.4 \times 65 = 780 \). Then, find the square root: \( s = \sqrt{780} \approx 27.93 \) Therefore, the speed of the car was approximately 27.93 mi/h.

Step 4: Rearrange Formula for Skid Distance in Part (b)

We need to find the distance \( d \) using the given speed \( s = 50 \) mi/h and for wet tar, where \( f = 0.5 \). Rearrange the formula to solve for \( d \): \( d = \frac{s^2}{30f} \).

Step 5: Calculate Skid Distance for Part (b)

Substitute the values into the rearranged formula: \[ d = \frac{50^2}{30 \times 0.5} \] Calculate \( 50^2 = 2500 \) and \( 30 \times 0.5 = 15 \). Thus, \( d = \frac{2500}{15} \approx 166.67 \) feet. Therefore, the car will skid approximately 166.67 feet.

Key Concepts

Skidding and Friction

When a car comes to a sudden stop, it may skid, or slide, along the road. This skidding happens because the wheels lose grip on the road surface. The distance a car skids can depend on how slippery the road is. The slipperiness is measured by something called the "coefficient of friction." The coefficient of friction (\( f \) ) is a number that tells us how much grip there is between the tires and the road. It changes based on the type of road surface and whether the surface is dry or wet. For example, a dry concrete road might have a high coefficient of friction (more grip). In contrast, a wet gravel road might have a low coefficient of friction (less grip).

• A higher coefficient means more friction, and usually, a shorter skid distance.
• A lower coefficient means less friction, leading to a longer skid distance.

Remember, the longer the skid marks, the higher the speed was likely to be.

Speed Calculation

To estimate how fast a car was going when it started skidding, we use a special formula: \[ s = \sqrt{30 f d} \]Here, \( s \) is the speed in miles per hour, \( f \) is the coefficient of friction, and \( d \) is the skid distance in feet. This formula helps police and investigators understand the situation after an accident by calculating the car's speed. Let's break it down:

• "30" is a constant. It helps match the units used in the formula.
• The term \( 30 f d \) inside the square root connects friction and skid distance to find speed.
• The square root (\( \sqrt{} \) ) means we need to find the number that, when multiplied by itself, equals \( 30 f d \).

This calculation helps ensure safety by showing how different driving conditions affect stopping distances.

Algebraic Formulas

Understanding algebraic formulas is crucial when working with physics problems like estimating speed or solving for skid distance. An algebraic formula is a mathematical equation that uses symbols to represent numbers and operations. The formula for estimating speed or finding skid distance involves rearranging terms to solve for a specific variable. For example, in Part (b) of the exercise, we rearranged the speed formula to calculate the skid distance. We started with the original speed formula and rearranged it to solve for \( d \), or skid distance:\[ d = \frac{s^2}{30f} \]This rearrangement process involves:

• Moving terms around the equation to isolate the target variable.
• Understanding that division and multiplication "undo" each other.
• Using algebra to transform a formula so you can find the unknown.

By practicing these steps, you become skilled in algebraic manipulation, helping you solve various physics-related problems in a straightforward manner.