The martian rover Curiosity can move across the landscape of Mars at speeds of up to \(4 \mathrm{cm} / \mathrm{s}\). In contrast, our typical walking speed is about \(4 \mathrm{km}\) per hour \((\mathrm{km} / \mathrm{h})\) a. How long would it take Curiosity to cross a soccer field \((110\) meters ) b. How long would it take you to walk the same distance?

Understanding and converting speed units is crucial in physics calculations. When converting units, it is essential to match them with the problem's requirements.

For instance, Curiosity's speed is given as 4 cm/s. We need this in meters per second (m/s) to maintain consistency with the distance given in meters. Remember that 1 meter = 100 centimeters. Therefore, to convert cm to m, divide by 100: \(4 \text{ cm/s} = \frac{4}{100} \text{ m/s} or 0.04 \text{ m/s}\).

Similarly, your walking speed is given in km/h. To convert km/h to m/s, first note that 1 km = 1000 meters and 1 hour = 3600 seconds. Using these conversions, we determine: \(4 \text{ km/h} = \frac{4 \times 1000 \text{ m}}{3600 \text{ s}} = \frac{4000 \text{ m}}{3600 \text{ s}} = \frac{10}{9} \text{ m/s}\).

Next, we look into calculating the time it takes to travel a certain distance. The formula to remember is:

\( \text{time} = \frac{\text{distance}}{\text{speed}}\)

This simple equation works because it divides the total distance by the speed, giving the total time in seconds (or the appropriate unit of time) needed to cover that distance.

Using Curiosity as an example, if it travels at a speed of 0.04 m/s, and the distance to be covered is 110 meters, the time taken would be:

\( \text{time}_{Curiosity} = \frac{110 \text{ m}}{0.04 { \text{ m/s}}}= 2750 \text{ s}\).

For walking, with a speed conversion of \(\frac{10}{9} \text{ m/s}\), the time to walk 110 meters would be:

\( \text{time}_{you} = \frac{110 \text{ m}}{\frac{10}{9} \text{ m/s}} = 110 \times \frac{9}{10} \text{ s} = 99 \text{ s}\).

By understanding and using the time calculation formula correctly, solving problems involving time becomes straightforward.

The distance formula plays a critical role in various physics problems, especially those involving speed and time.

The formula itself is a simple rearrangement of the speed equation:

\( \text{distance} = \text{speed} \times \text{time}\).

This is derived from the basic proportionality that distance covered is directly related to the speed and the time spent traveling. If you know two of the three variables (speed and time), you can always solve for the third.

For example, if Curiosity's speed is 0.04 m/s and it takes 2750 seconds to cross a field, the distance covered is:

\( \text{Distance}_{Curiosity} = 0.04 \text{ m/s} \times 2750 { \text{ s}}= 110 { \text{ m}}\).

Similarly, for you, traveling at \( \frac{10}{9} \text{ m/s}\) for 99 seconds:

\( \text{Distance}_{you} = \frac{10}{9} \text{ m/s} \times 99 \text{ s} = 110 \text{ m}\).

By mastering the distance formula, one can effortlessly navigate and solve various practical physics problems involving motion and travel.