Question

(a) If a car traveling 40 mph has a stopping distance of 112 it, find the stopping distance of a car that is traveling 60 mph.
(b) If it takes 7 ft for a car to stop, how fast was it traveling before the brakes were applied?

Answer 1

To solve these problems, we can use the relationship between speed and stopping distance. The stopping distance of a car is proportional to the square of its speed. This means that if you know the stopping distance at one speed, you can find the stopping distance at a different speed using the following formula:

\[
\frac{d_2}{d_1} = \left( \frac{v_2}{v_1} \right)^2
\]

where:
- \(d_1\) is the stopping distance at speed \(v_1\),
- \(d_2\) is the stopping distance at speed \(v_2\).

### (a) Finding the stopping distance at 60 mph

Given:
- \(v_1 = 40\) mph,
- \(d_1 = 112\) feet,
- \(v_2 = 60\) mph.

We will find \(d_2\), the stopping distance at 60 mph.

Using the proportionality:

\[
\frac{d_2}{112} = \left( \frac{60}{40} \right)^2
\]

Calculating \( \left( \frac{60}{40} \right)^2 \):

\[
\frac{60}{40} = 1.5
\]
\[
(1.5)^2 = 2.25
\]

Now substituting back:

\[
\frac{d_2}{112} = 2.25
\]

To find \(d_2\):

\[
d_2 = 112 \times 2.25 = 252 \text{ feet}
\]

So, the stopping distance of the car traveling at 60 mph is **252 feet**.

### (b) Finding the speed given a stopping distance of 7 ft

We want to find \(v_1\) given \(d_1 = 7\) feet, while using the stopping distance at 40 mph as our reference point.

We already know the proportionality:
- For 40 mph, \(d_1\) is 112 feet.

Now we want to find \(v_1\) corresponding to \(d_1 = 7\) feet. We use the relationship as before:

\[
\frac{7}{112} = \left( \frac{v_1}{40} \right)^2
\]

Calculating \( \frac{7}{112} \):

\[
\frac{7}{112} = \frac{1}{16}
\]

Now we can set up the equation:

\[
\frac{1}{16} = \left( \frac{v_1}{40} \right)^2
\]

Taking the square root of both sides gives us:

\[
\frac{1}{4} = \frac{v_1}{40}
\]

Solving for \(v_1\):

\[
v_1 = 40 \times \frac{1}{4} = 10 \text{ mph}
\]

Thus, the speed of the car before the brakes were applied was **10 mph**.