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Chapter 6: Problem 84

In January 2006, the New Horizons space probe was launched from Earth with the mission to perform a flyby study of Pluto. The arrival at the dwarf planet was estimated to happen after nine years, in 2015 . The distance between Earth and Pluto varies depending on the location of the planets in their orbits, but at their closest, the distance is 4.2 billion kilometers \((2.6\) billion miles). Calculate the minimum amount of time it takes for a transmitted signal from Pluto to reach the Earth.

Short Answer

Expert verified
The minimum amount of time it takes for a transmitted signal from Pluto to reach the Earth is approximately 3.89 hours.

Step by step solution

01

Find the speed of light in space

The speed of a transmitted signal in space is equal to the speed of light, which is approximately \(3.0 \times 10^8\) meters per second (m/s).
02

Convert distance to meters

The distance between Earth and Pluto at their closest is given as 4.2 billion kilometers. To use this value in our calculations, we need to convert it to meters. There are 1,000 meters in a kilometer, so we can find the distance in meters by multiplying the given distance by 1,000: \(4.2 \times 10^9\) kilometers × 1,000 meters/kilometer = \(4.2 \times 10^{12}\) meters.
03

Use the speed, distance, and time relationship

We know that the distance (d) traveled by an object can be expressed as the product of its speed (v) and the time (t) it takes to travel that distance: \(d = vt\). In this case, we want to find the time (t) taken for a signal to travel from Pluto to Earth, so we can rearrange the equation to solve for time: \(t = \frac{d}{v}\).
04

Plug in values and calculate time

Now we know the distance between Earth and Pluto (\(4.2 \times 10^{12}\) meters) and the speed of light (\(3.0 \times 10^8\) meters per second), so we can plug these values into the formula we derived above: \(t = \frac{4.2 \times 10^{12}\,\text{meters}}{3.0 \times 10^8\,\text{meters/s}}\).
05

Simplify and find the answer

Divide the given values to find the time: \(t = \frac{4.2 \times 10^{12}}{3.0 \times 10^8} = 14 \times 10^3\) seconds. Now, we could convert this to minutes by dividing by 60 seconds per minute: \(14 \times 10^3\,\text{seconds} ÷ 60\,\text{seconds/minute} \approx 233.33\) minutes. To get a more intuitive value, we can convert this to hours by dividing by 60 minutes per hour: \(233.33\,\text{minutes} ÷ 60\,\text{minutes/hour} \approx 3.89\) hours. So, the minimum amount of time it takes for a transmitted signal from Pluto to reach the Earth is approximately 3.89 hours.

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Most popular questions from this chapter

Determine which of the following statements are false and correct them. (a) The frequency of radiation increases as the wavelength increases. (b) Electromagnetic radiation travels through a vacuum at a constant speed, regardless of wavelength. (c) Infrared light has higher frequencies than visible light. (d) The glow from a fireplace, the energy within a microwave oven, and a foghorn blast are all forms of electromagnetic radiation.

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