Question

An asteroid traveling at 20 km/s collides with Earth causing a slowdown of its rotational motion. Following the impact, the day is lengthened by one second. Before the impact, we consider the moment of inertia of the asteroid as that of a point mass. After the impact, the disintegration of the asteroid is complete and all its mass is deposited evenly over the entire surface of the earth forming a holow sphere with a radius equivalent to that of the earth. Thus, the moment of inertia of the asteroid can no longer be considered to be that of a point mass. The Earth and the asteroid are solid spheres. If the asteroid has the same density as that of the Earth, determine the radius that the asteroid should have if it strikes the earth radially,

Answer 1

The radius of an asteroid that would cause Earth's day to lengthen by one second upon impact can be determined using conservation of angular momentum, given the mass, density, and impact velocity of the asteroid, along with Earth's rotational parameters.

Step-by-step explanation:

The question asks about the change in Earth's rotational motion after being struck by an asteroid, specifically how to determine the radius of an asteroid that would cause a day to be lengthened by one second after hitting Earth radially. Since the asteroid's density is the same as Earth's and it disintegrates to form a hollow sphere over the Earth after impact, conservation of angular momentum can be used to solve for the asteroid's radius given Earth's moment of inertia, the rotational speed decrease, and the asteroid's impact velocity.

When the asteroid hits Earth, the angular momentum is conserved. Before impact, Earth's angular momentum can be calculated using its mass, radius, and angular velocity. After impact, the system's new moment of inertia includes both Earth's and the disintegrated asteroid's mass distributed evenly on Earth's surface. The rotational speed change is accounted for, given that a day increases by one second. By equating initial and final angular momentums, solving for the mass of the asteroid, and then utilizing its density, we can find its radius.