For any two objects that travel in space, it is important to know whether they c

Question

For any two objects that travel in space, it is important to know whether they collide. If two objects collide, they must satisfy the following: i. Their trajectories meet with each other; ii. The reach some intersection point of their trajectories at the same time. Now we assume that our earth is rotating around the sun with the position r(t) = (R sin(2 pi t), R cos(2 pi t),0) at time t. Here R is the distance from the earth to the sun and t is counted in years. A satellite just warns us that a very dangerous asteroid is traveling exactly on the same xy-plane as the earth, whose position is (2 pi Rt, 4 pi^2 Rt^2 - 1/2 R, 0) at time t. (a) Do the trajectories of the earth and the asteroid meet with each other? Explain. (b) Will the earth and the asteroid collide? Explain.

Explanation / Answer

if they collide there x and y coordinate will be same at time t ie

Comparing the x cordinate

R*sin(2*pi*t) = 2*pi*R*t => Sin(2*pi*t) = 2*pi*t => t = 0 there is no other value for this the equn will true

Comparing y corodinate

R*cos(2*pi*t) = 4*pi^2*R*t^2 -R/2

if we put t=0 we get

R = - R/2

it is never possible as R is not zero distance between sun & earth

So they never collide and their trajectory never meet.

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.