Question

A speed of \(7 \mathrm{~mm} / \mu \mathrm{s}\) is equal to _____ . a) \(7000 \mathrm{~m} / \mathrm{s}\) b) \(70 \mathrm{~m} / \mathrm{s}\) c) \(7 \mathrm{~m} / \mathrm{s}\) d) \(0.07 \mathrm{~m} / \mathrm{s}\)

Short answer

Answer: a) 7000 m/s

Step-by-step solution

Write down the given speed

The given speed is \(7\mathrm{~mm}/\mu\mathrm{s}\).

Convert millimeters to meters

We know that \(1\mathrm{~m} = 1000\mathrm{~mm}\). So, we can divide by 1000 to convert the millimeters to meters: $$\frac{7\mathrm{~mm}}{1\mu\mathrm{s}} × \frac{1\mathrm{~m}}{1000\mathrm{~mm}} = \frac{7}{1000}\mathrm{~m}/\mu\mathrm{s} = 0.007\mathrm{~m}/\mu\mathrm{s}$$

Convert microseconds to seconds

We know that \(1\mathrm{~s} = 10^6\mu\mathrm{s}\). So, we can multiply by \(10^6\) to convert the microseconds to seconds: $$0.007\mathrm{~m}/\mu\mathrm{s} × \frac{1\mathrm{~s}}{10^6\mu\mathrm{s}} = 0.007 × 10^6\mathrm{~m}/\mathrm{s} = 7000\mathrm{~m}/\mathrm{s}$$

Identify the correct answer

Based on our calculations, a speed of \(7\mathrm{~mm}/\mu\mathrm{s}\) is equal to \(7000\mathrm{~m}/\mathrm{s}\). Therefore, the correct answer is: a) \(7000\mathrm{~m}/\mathrm{s}\).

Key concepts

Speed Unit Conversion

Understanding speed unit conversion is crucial in physics as it allows us to compare and work with speeds in different measurement systems. In the case of the exercise, we're concerned with converting a speed given in millimeters per microsecond (mm/µs) to meters per second (m/s).

Amongst the most common speed conversions is changing from smaller units to larger units, as seen with mm to m, where we must remember that 1 m equals 1000 mm. Similarly, the time units are converted from microseconds to seconds, knowing that 1 second is comprised of 1,000,000 microseconds. The ability to nimbly shift between these units can prove crucial in fields like material science, where small distances and extremely short time intervals matter.

For our example, the speed of 7 mm/µs converts to a much larger number when expressed in m/s because we have changed to larger distance and time units. This highlights a basic principle in unit conversion: if you convert to a larger unit of measure, the numerical value decreases, but if you convert to a larger time unit, the speed value increases.

Millimeters to Meters Conversion

One of the fundamental concepts in physics is the conversion between different units of length. In this exercise, we converted millimeters (mm) to meters (m), two units of length in the metric system. The key to converting units like these is understanding the relationship between them: 1 m is equal to 1000 mm.

To perform this conversion, you simply divide the number of millimeters by 1000. It helps to remember that 'milli-' means one-thousandth, so there are 1000 millimeters in a meter.

Why is this important?

The ability to convert between millimeters and meters can be critical in numerous scientific calculations where precision is required. Whether you're measuring the thickness of a biological cell or the length of a sports field, understanding how to accurately move between different scales of measurement is essential in accurately interpreting and communicating data.

Microseconds to Seconds Conversion

Just as with the conversion of length units, time units also have established relationships that are essential for unit conversions in physics. Converting microseconds (µs) to seconds (s) is straightforward once you remember that 1 second equals 1,000,000 microseconds. The prefix 'micro-' signifies a millionth part of a unit.

To convert microseconds to seconds, you divide the number of microseconds by 1,000,000. This type of conversion is particularly useful in measuring time intervals in electronics and communications, where events occur within fractions of a second.

Real-World Application

In practice, handling conversions between microseconds and seconds is imperative in fields like computer science, where processor speeds are often measured in nanoseconds or microseconds, and timing is critical for operation sequences. This highlights the importance of mastering such conversions to better understand and work with dynamic systems that hinge on precise time measurements.