Video: Determining the Distance Traveled at a Constant Speed

Voyager 1 is the farthest man-made object from Earth. A radio signal from Earth takes 20.6 hours to reach Voyager 1. If the speed of the radio signal is 3 × 10⁸ m/s, calculate how far away Voyager 1 is at the instant that it receives the signal. Give your answer to 2 decimal places. [A] 2.22 × 10¹³ m [B] 1.25 × 10¹⁵ m [C] 6.18 × 10⁹ m [D] 3.71 × 10¹¹ m

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Video Transcript

Voyager 1 is the farthest man-made object from Earth. A radio signal from Earth takes 20.6 hours to reach Voyager 1. If the speed of the radio signal is three times 10 to the 8 meters per second, calculate how far away Voyager 1 is at the instant that it receives the signal. Give your answer to two decimal places. (A) 2.22 times 10 to the 13 meters. (B) 1.25 times 10 to the 15 meters. (C) 6.18 times 10 to the nine meters. (D) 3.71 times 10 to the 11 meters.

Okay, so in this question, we have a radio signal that is being sent from Earth to Voyager 1. We are told that the time it takes to get there is 20.6 hours and that the speed of the radio signal is three times 10 to the eight meters per second. We are then asked to work out how far away Voyager 1 is when it receives the signal. In other words, the distance between Voyager 1 and Earth. The question has given us a value of time which we’ll label 𝑡. So we have that 𝑡 is equal to 20.6 hours. We are also given a value of speed which we’ll label 𝑠. So we have 𝑠 equals three times 10 to the eight meters per second. We are looking to find a value of distance which we’ll label 𝑑.

We should recall that there is a formula that relates these three quantities. This formula says that speed 𝑠 is equal to distance 𝑑 divided by time 𝑡. We are looking to find the value of 𝑑, the distance, so let’s rearrange this formula to make 𝑑 the subject. If we multiply both sides of the equation by 𝑡, then we see that on the right-hand side, the 𝑡’s in the numerator and denominator cancel each other out. Then swapping the left- and right-hand sides of the equation over, we have that distance 𝑑 is equal to speed 𝑠 multiplied by time 𝑡. We have values for both 𝑠 and 𝑡. But in order to use this formula to calculate 𝑑, we need our quantities to have consistent units.

Our speed is measured in units of meters per second, while our time is measured in hours. So, before we can use these quantities in our formula for the distance, we need to convert our time into units of seconds. We can recall that there are 60 minutes in one hour and 60 seconds in one minute. So, to convert from units of hours to units of seconds, we take our value of time measured in hours and we multiply it by 60 minutes per hour and then multiply again by 60 seconds per minute. If we track what’s going on with the units here, we see that the hours cancel with the per hour and the minutes cancel with the per minute. Then we are left with units of seconds.

When we do the multiplication, we find that the value of the time 𝑡 in seconds is equal to 74,160 seconds. We can write that time in scientific notation as 7.416 times 10 to the four seconds. Now that we have the speed 𝑠 and the time 𝑡 in consistent units, we can substitute these into our formula to calculate the value of 𝑑. When we do this, we have that 𝑑 is equal to three times 10 to the eight meters per second multiplied by 7.416 times 10 to the four seconds. Doing this multiplication gives us a result of 2.2248 times 10 to the 13 meters. And this value is the distance 𝑑 of Voyager 1 from Earth at the instant that it receives the radio signal. So this is what the question was asking us to work out.

But if we look back at the question, we see that we we were asked to give our answer to two decimal places. To two decimal places, our value of 𝑑 rounds down to 2.22 times 10 to the 13 meters. This matches the value given here in option (A). So our final answer to the question is that the distance of Voyager 1 from Earth at the time at which it receives the radio signal is given here by option (A), 2.22 times 10 to the 13 meters.

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