QuestionCalculate the distance from Uranus to Earth in km, given light speed $3 \times 10^{8} \mathrm{~m/s}$ and time 0.105 day.

Studdy Solution

STEP 1

Assumptions1. The speed of light is $3 \times10^{8} \mathrm{~m/s}$ . The time required by the light from Uranus to reach the Earth is0.105 day3. We want to find the distance in kilometers and express the answer in scientific notation4. We assume that the speed of light is constant and that it travels in a straight line5. We know that distance is equal to speed multiplied by time

STEP 2

First, we need to convert the time from days to seconds because the speed of light is given in meters per second.
$Time =0.105\, days \times24\, hours/day \times60\, minutes/hour \times60\, seconds/minute$

STEP 3

Now, calculate the time in seconds.
$Time =0.105 \times24 \times60 \times60 =9072\, seconds$

STEP 4

Now that we have the time in the same units as the speed of light, we can calculate the distance light travels from Uranus to Earth using the formula for distance, which is speed multiplied by time.
$Distance = Speed \times Time$

STEP 5

Plug in the values for the speed of light and the time to calculate the distance.
$Distance =3 \times10^{8}\, m/s \times9072\, s$

STEP 6

Calculate the distance in meters.
$Distance =3 \times10^{8} \times9072 =2.7216 \times10^{12}\, m$

STEP 7

Now, we need to convert the distance from meters to kilometers because the problem asks for the distance in kilometers. We know that1 kilometer is equal to1000 meters.
$Distance =2.7216 \times10^{12}\, m \times1\, km/1000\, m$

STEP 8

Calculate the distance in kilometers.
$Distance =2.7216 \times10^{12} /1000 =2.7216 \times10^{}\, km$

STEP 9

Finally, we need to express the answer in scientific notation with3 significant figures. This means we need to round the number to the nearest hundred million.
$Distance =2.72 \times^{9}\, km$ The distance between Uranus and the Earth is $2.72 \times^{9}$ kilometers.

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