Question

An astronomer on the Earth observes a meteoroid in the southern sky approaching the Earth at a speed of 0.800c. At the time of its discovery the meteoroid is 20.0 ly from the Earth. (8 points 1 light-year (ly) is a unit of length used to express astronomical distances and is equivalent to about 9.46 trillion kilometres (9.46x10¹² km). Calculate the time interval required for the meteoroid to reach the Earth as measured by the Earthbound astronomer,

Answer 1

The time interval required for a meteoroid traveling at 0.800c to cover a distance of 20.0 light-years is calculated by dividing the distance by the meteoroid's speed. This results in a time of 25 years for the meteoroid to reach Earth from the perspective of an Earthbound astronomer.

Step-by-step explanation:

The student's question is about the time it would take for a meteoroid traveling at a speed of 0.800c to reach Earth from a distance of 20.0 light-years (ly) as measured by an Earthbound astronomer. Since one light-year is the distance that light travels in one year, and since we know the speed of the meteoroid in relation to the speed of light (c), we can calculate the time interval for its journey. We simply divide the total distance (20.0 ly) by the speed of the meteoroid (0.800c)

To calculate this, consider that light travels at approximately 300,000 kilometers per second and thus covers a distance of one light-year in one year. Since the meteoroid is traveling at 0.800c, it will cover the distance of 20.0 light-years in a time given by time = distance/speed = 20.0 ly / 0.800c. To find the time in years, we know that c is the speed of light, hence 1c is the time light takes to travel 1 ly, which is one year. Therefore, the meteoroid will take 20.0 ly / 0.800 or 25 years to reach Earth, as measured by the Earthbound astronomer.

Answer 2

The time interval required for the meteoroid to reach the Earth as measured by the Earthbound astronomer is
`1.31*10^9\ s`

How to calculate the time interval?

First, we shall calculate the time experienced by the meteoroid. Details below:

• Speed of light (c) = 3×10⁸ m/s
• Speed of meteoroid (v) = 0800c = 0.8 × 3×10⁸ = 2.4×10⁸ m/s
• Distance traveled by (d) = 20 light year = 20 × 1000 × 9.46×10¹² = 189.2×10¹⁵ m
• Time by meteoroid (t) = ?

t = d / v

= 189.2×10¹⁵ / 2.4×10⁸

= 7.88×10⁸ s

Finally, we shall calculate the time interval for the meteoroid to reach the Earth. Details below:

• Time by meteoroid (t) = 7.88×10⁸ s
• Time interval required to reach earth (Δt) =?

`\Delta t = \frac{t}{\sqrt{1 \ -\ (v^2)/(c^2)}} \\\\\Delta t = \frac{t}{\sqrt{1 \ -\ ((0.8c)^2)/(c^2)}} \\\\\Delta t = (t)/(√(1 \ -\ 0.8^2)) \\\\\Delta t = (7.88*10^(8))/(√(1 \ -\ 0.8^2))\\\\\Delta t = 1.31*10^9\ s`