A car is traveling on a curve that forms a circular arc. The force F needed to keep the car from skidding is jointly proportional to the weight $\text {W }$
Of the car and the square of its speed s, and is inversely proportional to the radius r of the curve. A car weighing 2,500 lbs travels around a curve at 70 mi/h. The next car to round this curve weighs 4,900 lbs and requires the same force as the first car to keep from skidding. How fast is the second car traveling?
A) $49 \mathrm { mi } / \mathrm { h }$
B) $50 \mathrm { mi } / \mathrm { h }$
C) $44 \mathrm { mi } / \mathrm { h }$
D) $54 \mathrm {~min}$
E) $52 \mathrm { mi } / \mathrm { h }$

Question

A car is traveling on a curve that forms a circular arc. The force F needed to keep the car from skidding is jointly proportional to the weight $\text {W }$
Of the car and the square of its speed s, and is inversely proportional to the radius r of the curve. A car weighing 2,500 lbs travels around a curve at 70 mi/h. The next car to round this curve weighs 4,900 lbs and requires the same force as the first car to keep from skidding. How fast is the second car traveling? 
A) $49 \mathrm { mi } / \mathrm { h }$ 
B) $50 \mathrm { mi } / \mathrm { h }$ 
C) $44 \mathrm { mi } / \mathrm { h }$ 
D) $54 \mathrm {~min}$ 
E) $52 \mathrm { mi } / \mathrm { h }$

Quizplus · Accepted Answer

Using the given information, we can set up a proportion as follows:

$$\frac{F_1}{F_2} = \frac{W_1s_1^2}{W_2s_2^2} \cdot \frac{r_2}{r_1}$$

where subscripts 1 and 2 refer to the first and second car, respectively.

We know that $F_1=F_2$ and $W_1=2500$, so we can simplify the proportion to:

$$s_1^2 = \frac{W_2}{W_1} \cdot \frac{r_1}{r_2} \cdot s_2^2$$

Plugging in the given values for the first car, we have:

$$70^2 = \frac{4900}{2500} \cdot r_1 \cdot s_2^2$$

Solving for $r_1$, we get:

$$r_1 = \frac{70^2 \cdot 2500}{4900 \cdot s_2^2} = \frac{245}{2s_2^2}$$

Now we can use the given information about the second car to solve for its speed $s_2$:

$$F_2 = k \cdot W_2 \cdot s_2^2 \cdot \frac{1}{r_1} = F_1 = k \cdot W_1 \cdot s_1^2 \cdot \frac{1}{r_1}$$

Plugging in the values for the first car and cancelling out $r_1$ on both sides, we get:

$$k \cdot 2500 \cdot 70^2 = k \cdot 4900 \cdot s_2^2 \cdot \frac{245}{2s_2^2}$$

Simplifying and solving for $s_2$, we get:

$$s_2 = \sqrt{\frac{2500 \cdot 70^2}{2 \cdot 4900 \cdot 245}} \approx 50 \mathrm{~mi/h}$$

Therefore, the answer is (B) $50~\mathrm{mi/h}$.

A Car Is Traveling on a Curve That Forms a Circular