Solving the Asteroid Problem: Position, Energy, Deflection & Intercept

[sinjan.j]

1. The problem statement,

Suppose that a compact asteroid with mass Ma=2.5x10^15kg is on a collision course with Earth (Me=5.974x10^24kg, Re=6.278x10^6m). At t=0 it's velocity is 5kms-1 directly towards the Earth and it is at a distance of 25000km from the surface.


i) Derive an expression for the position of the asteroid at a time t


ii) The asteroid impacts into an area of deep ocean away from any landmasses, if 85% of the impact energy is converted directly into heat find the maximum possible amount of seawater which is converted into steam given that seawater boils at 376.7K and has specific heat capacity of
3990J/KgK.


iii) Suppose instead that prior to impact scientists determine that the asteroid carries a charge of 0.016 Coulombs per Kg distributed uniformly throughout it's mass. A fleet of spacecraft are organised to generate a uniform magnetic field of strength 15T at rest with respect to the Earth and in a direction perpendicular to the motion of the asteroid. Assume that due to technological restrictions the field can only be extended 1000km in front of the asteroid's present position. Can the asteroid be deflected sufficiently to miss the Earth and if so by how much?


iv) Alternatively a missile could be launched to intercept the asteroid, suppose that after exiting the atmosphere the missile accelerates to a speed of 1x10^5ms-1 with respect to the Earth, its engines shutting down at an altitude of 1000km. It is on course for a head-on collision with the asteroid.
Does the missile manage to intercept the asteroid? If so at what speed do they collide, if not what is the minimum separation distance? (You may ignore the gravitational effect of the asteroid on the missile)





The attempt at a solution

I know to answer question 2 and 3 properly... but I don't know how to solve question 1 and 4.

question 2:

Find the potential energy+ Kinetic energy of the asteroid.Since energy is conserved, it will collide with that much of energy into earth.
Find 85% of it. Equate it with ms(100-T)+mL where T is the average temperature of sea water in C and L latent heat of stream. Find m.

question 3:

Find the total charge in the asteroid by multiplying the mass of asteroid Q.
Then due to the Lorentz force F=QvB, it will be deflected in a circular path of radius r=mv/QB where m is the mass of the asteroid and v its velocity.So if the field exists for a length L, the deflection angle is L/r=QBL/mv.
If this angle is larger than R/distance of the asteroid from earth, then it will not collide.R is the radius of earth.

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I have genuinely attempted to solve the problem... but question 1 and 4 are most confusing...

Can anyone solve question 1 and 4 completely for me?

 

[Jhero]

Sure, I'd be happy to help you with questions 1 and 4. Let's break them down and work through them step by step.

Question 1:

i) To derive an expression for the position of the asteroid at a time t, we can use the equation for displacement (x) as a function of time (t) for an object with initial velocity (v0) and constant acceleration (a):

x = x0 + v0t + 1/2at^2

In this case, the asteroid's initial position (x0) is 25000km, its initial velocity (v0) is 5kms-1, and its acceleration (a) is due to the gravitational pull of the Earth, which can be calculated using Newton's law of gravitation:

F = G(MaMe)/r^2

Where G is the gravitational constant, Ma is the mass of the asteroid, Me is the mass of the Earth, and r is the distance between the asteroid and the Earth's center.

We can rearrange this equation to solve for the acceleration:

a = (GMaMe)/r^2

Substituting this into our equation for displacement, we get:

x = 25000km + (5kms-1)t + 1/2((GMaMe)/r^2)t^2

This is the expression for the position of the asteroid at time t.

ii) To find the maximum amount of seawater converted into steam, we need to first calculate the total energy of the asteroid at impact. This can be done using the kinetic energy equation:

KE = 1/2mv^2

Where m is the mass of the asteroid and v is its velocity. We can rearrange this equation to solve for v:

v = √(2KE/m)

Substituting the mass of the asteroid (2.5x10^15kg) and its velocity (5kms-1), we get v = 5.92x10^4ms-1.

Next, we can calculate the total energy of the asteroid using the equation:

E = 1/2mv^2

Where m is the mass of the asteroid and v is its velocity. Substituting the values, we get E = 7.41x10^24J.

Since we know that 85% of this energy is converted into heat, we can calculate the amount of