russ_watters said:
It is of course possible. But that doesn't change the issue. An object that was just barely overtaking Earth (or vice versa) would still accelerate to (just barely above) escape velocity at impact.
So let's say the Earth is approaching perigee. It's speed with no orbital perturbations would be:
[tex]v=\sqrt{(\mu_h (\frac{2}{147.1x10^6 km} - \frac{1}{149.6x10^6 km})}=30.285 km/sec[/tex]
heliocentric gravitational constant is 1.327x10^11 km^3/sec^2
semi-major axis of Earth's orbit is 149.6 million kilometers
perigee is 147.1 million kilometers
You have an asteroid with a semi-major axis of 145 million kilometers and its apogee is 147.2 million kilometers, meaning its speed is:
[tex]v=\sqrt{(\mu_h (\frac{2}{147.2x10^6 km} - \frac{1}{145x10^6 km})}=29.8 km/sec[/tex]
The asteroid goes slightly further out from the Sun than the Earth and the Earth is overtaking it. The relative velocity between the two is 485 meters/sec (0.485 km/sec)
The asteroid's gravity adds a bit of energy to the Earth's orbit, while the Earth subtracts quite a bit of energy from the asteroid's orbit.
You're going to get into issues as to the directions of the velocity vectors, making this a tough calculation, however, it can be made easier by the calculating the specific energy due to gravitational attraction.
[tex]\epsilon=\frac{v^2}{2}-\frac{\mu_g}{d}[/tex]
with the geocentric gravitational constant being 3.986x10^6 km^3/sec^2
When the specific energy is 0, the object has achieved escape velocity for that current position (11 km/sec is escape velocity at the surface of the Earth, which is actually a really bizarre place to calculate it given the problems of traveling 11 km/sec through the Earth's atmosphere).
If the Earth and the asteroid are within 3.4 million km when the asteroid is at apogee and the Earth is at perigee, then the asteroid will not achieve escape trajectory relative to the Earth. If the Earth and the asteroid are at least 3.4 million km apart, then the asteroid will achieve escape velocity relative to the Earth, but could still impact the Earth just because of the direction of its trajectory. (3.4 million sounds big, but that's actually about 1.3 degrees of the Earth's orbit.)