Question

How far does your car, moving at \(70 \mathrm{mi} / \mathrm{h}(=112 \mathrm{~km} / \mathrm{h})\) travel forward during the \(1 \mathrm{~s}\) of time that you take to look at an accident on the side of the road?

Short answer

During the 1 second that a driver takes to look at an accident on the side of the road, the car (moving at 70 mph) will travel approximately \(0.019444 \) miles forward.

Step-by-step solution

Understanding the problem

The important piece of information in this exercise is that the car is moving at a speed of 70 mph (or 112 km/h). We want to find out how far the car travels in 1 second. The unit of speed given (miles or kilometers per hour) is not compatible with the time given (in seconds), so we'll need to do a unit conversion.

Converting Units

Let's convert the speed from mph to miles per second. We know that 1 hour is equal to 3600 seconds. Therefore, we divide the speed by 3600 to get the speed in miles per second, \(70 \, mph \,/\, 3600 = 0.019444 \, miles/s\). Now the units of speed and time are compatible.

Calculating Distance

We know that distance = speed x time. So we multiply the speed in miles per second (0.019444 miles/s) by the time (1 second) to find out how far the car travels in that time. \(Distance = 0.019444 \, miles/s \times 1 \, s = 0.019444 \, miles\)

Key concepts

Speed Calculation

Speed calculation is the key to understanding how fast something is moving. It determines the rate at which an object covers distance over time. In our example, the car is moving at a speed of 70 miles per hour. Of course, understanding speed in miles per hour can be common in everyday scenarios, but it is crucial to remember this is only applicable when the time is factored in hours.

To calculate speed properly, you typically need two components:

• Distance traveled - in this case, miles
• Time taken - here it's shown in hours

These components together give you the average speed using the formula:\[\text{Speed} = \frac{\text{Distance}}{\text{Time}}\]In scenarios requiring different units, like our example where time is in seconds, you need a unit conversion, which ensures that calculations remain accurate and meaningful.

Distance Calculation

The purpose of distance calculation is to find out how far an object travels. With the speed known, and the time taken to cover the distance given, you can calculate the distance. This comes down to a straightforward equation of using the speed and time to find out the travel length.

The famous formula for calculating distance is:\[\text{Distance} = \text{Speed} \times \text{Time}\]For the exercise, we've already converted the speed from miles per hour to miles per second, resulting in \(0.019444\) miles per second.

By multiplying this speed by the 1 second of time provided, the distance calculation becomes simple. So, in this problem, the car travels a short distance of \(0.019444\) miles.

Time Conversion

Time conversion is crucial when dealing with speed and distance calculations, as it ensures all units align correctly. In our situation, the initial unit of speed is in hours, whereas time given is in seconds. Thus, conversion clears any discrepancy.

A straightforward conversion is needed:- Convert hours to seconds to match the time unit given in seconds.We know there are 3600 seconds in an hour, so it's necessary to adjust the speed from miles per hour to miles per second by dividing by 3600:
\[70 \, mp/h \div 3600 = 0.019444 \, miles/second\]Now, with speed and time both in compatible units, calculations are much simpler and more precise whether they involve distance traveled or determining speed over different time intervals.