INT Most geologists believe that the dinosaurs became extinct 65 million years ago when a large comet or asteroid struck the earth, throwing up so much dust that the sun was blocked out for a period of many months. Suppose an asteroid with a diameter of 2.0 km and a mass of 1.0×10¹³ kg hits the earth (6.0×10²⁴ kg) with an impact speed of 4.0×10⁴ m/s.
b. What percentage is this of the earth's speed around the sun? The earth orbits the sun at a distance of 1.5×10¹¹ m .

Question

Accepted Answer

Hey, everyone. So this problem is dealing with conservation of momentum. Let's see what it's asking us. We have a giant comet of 1.5 kilometer radius with a mass of 3.2 times 10 of the 14 kg slams into the planet. Jupiter at a speed of 7.5 times 10 to the 5 m per second. Jupiter recoils at this speed. V sub R following this collision, we are asked to find the ratio of VR divided by VJ where VJ is Jupiter speed in its orbit around the sun. We're told to assume that this collision is in elastic. And then they give us some astronomical data where we have Jupiter's mass. Jupiter's orbital period, Jupiter's orbital period and the average distance from the sun that we will um use and to solve the problem. So our multiple choice answers here are a 8.4 times 10 to the negative 13 B 9.6 times 10 to the negative 12 C 6.1 times 10 to the negative 11 or D 4.8 times 10 to the negative 10. And so when we are told that the collision is in elastic. It's a hint that we can use the conservation of momentum where we have our initial momentum is equal to our final momentum. And momentum is given by the equation mass multiplied by velocity. So initially, we have the mass of Jupiter multiplied by the initial speed of Jupiter plus the mass of the comet multiplied by the initial speed of the comet is equal to in the final scenario. After the collision, the masses are combined because of this in elastic collision. So mass of Jupiter plus mass of the comet all multiplied by the final velocity. So looking at these terms, our initial velocity of Jupiter is zero because we are looking at our initial velocity of um of the comet with respect to Jupiter, not with Jupiter is in respect to the sun. So this term goes to zero and then solving for VF which we are told is VRV seven R, we get the mass of the comet or 3.2 times 10 to the 14 kg multiplied by the initial speed of the comet or 7.5 times 10 to the fifth meters per second, divided by the mass of Jupiter, which was given to us in the problem as 1.898 times 10 to the 27 kg plus the mass of the comet or 3.2 times 10 to the 14 kg. And so that gives us a visa R of 1.26 times 10 to the negative 7 m per second. However, what they're asking us to find is the ratio of this V sub R divided by V sub J where Visa Js Jupiter speed um R in during its orbit around the sun. So in relation to the sun and so this VR divided by VJ, we can use our uniform circular motion equation which we can recall is given by V equals two pi multiplied by R divided by T. And so the R is the radius of Jupiter that's given to us in the problem. Um or sorry, not the radius of Jupiter, the radius of the orbit, which is the distance between Jupiter and the sun. And that's given to us in the problem. The 7.78 times 10 to the watching units, 10 to the eighth kilometers is 10 to the 11th meters. We wanna keep everything in standard units and then we are going to divide by our period T which here is 11.86 years. And so we need to convert that to seconds. So we multiply that by 365 days per year, 24 hours a day and 3600 seconds per hour. And so when we plug that in to our calculator, we get a speed of Jupiter as it orbits the sun of 1.31 times 10 to the 4 m per second. And then our last step here is finding this ratio VR divided by VJ. And we simply have to divide those um two speeds that we found. So 1.26 times 10 to the negative 7 m per second, divided by 1.31 times 10 to the 4 m per second. And that gives us 9.6 times 10 to the negative 12. So that is the correct answer for this problem. And when we look at our multiple choice answers that aligns with answer choice B so that's all we have for this one. We'll see you in the next video.

Verified Solution

Key Concepts

Kinetic Energy

Orbital Speed

Percentage Calculation