The formula you used, Ep + Ek = Ek + Ep, is not the correct formula for this scenario. The formula you need to use is the conservation of energy equation, which states that the total energy (potential energy + kinetic energy) of a system remains constant. In this case, the system is the asteroid and the Earth.
The formula for conservation of energy is:
E = Ep + Ek = constant
Where:
E = total energy
Ep = potential energy
Ek = kinetic energy
In this problem, we can assume that the asteroid has no initial potential energy, as it is far away from the Earth. Therefore, the equation becomes:
E = Ek = constant
Using this equation, we can solve for the final kinetic energy of the asteroid just before impact. The final kinetic energy will be equal to the initial potential energy, which is given by the formula you mentioned, Ep = GMm/r.
So, we can write:
Ek = GMm/r
Now, we can plug in the values given in the problem:
Ek = (6.67×10^-11 Nm^2/kg^2)(5.98×10^24 kg)(5.05×10^9 kg)/(6.38×10^6 m) = 5.01×10^15 J
Since we now have the final kinetic energy of the asteroid, we can use the formula for kinetic energy, Ek = 1/2mv^2, to solve for the final velocity (v) of the asteroid just before impact.
Rearranging the formula, we get:
v = √(2Ek/m)
Plugging in the values, we get:
v = √(2(5.01×10^15 J)/(5.05×10^9 kg)) = 6.26×10^3 m/s
Therefore, the asteroid will hit the Earth's surface with a speed of approximately 6.26 km/s. It is important to note that this is a theoretical calculation and does not take into account the effects of atmospheric friction, which would likely decrease the speed of the asteroid slightly. Nevertheless, it is clear that the asteroid will hit the Earth with a tremendous amount of energy and could potentially cause significant damage. It is important for NASA to continue monitoring the asteroid and come up with a plan to prevent a collision with Earth.
