So this is the problem:

A comet is in an elliptical orbit around our Sun. At its point of closest approach, it is moving at 6.3x10^4 m/s and is 4.2x10^10m away from the sun. When it is at the farthest distance, 7.5x10^12m away, how fast is it going?

This is what I did

K1+U1=K2+U2 (since energy should be conserved)

so...

1/2m1v1^2 + -Gm1m2/r1 = 1/2m1v2^2 + -Gm1m2/r2

the m1 cancel since all of them have it

1/2v1^2 + -Gm2/r1 = 1/2v2^2 + -Gm2/r2

then I rearranged to get v2 by itself

1/2v1^2 + -Gm2/r1 + Gm2/r2 = 1/2v2^2

*2 *2

(v1)^2 +-2Gm2/r1 + 2Gm2/r2 = (v2)^2

(v1)^2 + 2Gm2/r2 -2Gm2/r1 = (v2)^2

simplified it

(v1)^2 + (2Gm2 (1/r2 - 1/r1))= (v2)^2

and then square root for the final equation to be

((v1)^2 + (2Gm2 (1/r2 - 1/r1)))^.5 = (v2)

Then I plugged in the numbers

v1 = 6.3x10^4
G= 6.67x10^-11
r2= 7.5x10^12
r1= 4.2x10^10
m2 (mass sun) = 2x10^30

But when I plugged all this in to solve for v2 I got a negative number under the square root. Where did I go wrong?

To determine where you went wrong, let's go through the steps and calculations to identify any mistakes.

1. Start with the conservation of energy equation: K1 + U1 = K2 + U2.

2. Simplify the equation using the given information:
- Kinetic energy at closest approach: 1/2 * (6.3 × 10^4)^2.
- Gravitational potential energy at closest approach: -G * (2 × 10^30) * (6.3 × 10^4) / (4.2 × 10^10).
- Gravitational potential energy at farthest distance: -G * (2 × 10^30) * (6.3 × 10^4) / (7.5 × 10^12).
- Kinetic energy at farthest distance: 1/2 * v2^2.

3. Substitute these values into the equation:
1/2 * (6.3 × 10^4)^2 - G * (2 × 10^30) * (6.3 × 10^4) / (4.2 × 10^10) = 1/2 * v2^2 - G * (2 × 10^30) * (6.3 × 10^4) / (7.5 × 10^12).

4. Rearranging the equation to solve for v2:
1/2 * v2^2 = 1/2 * (6.3 × 10^4)^2 - G * (2 × 10^30) * (6.3 × 10^4) / (4.2 × 10^10) + G * (2 × 10^30) * (6.3 × 10^4) / (7.5 × 10^12).

5. Simplify and evaluate the equation:
1/2 * v2^2 = 1/2 * (6.3 × 10^4)^2 + G * (2 × 10^30) * (6.3 × 10^4) * (1 / (7.5 × 10^12) - 1 / (4.2 × 10^10)).

6. Substitute the numerical values:
1/2 * v2^2 = 1/2 * (6.3 × 10^4)^2 + (6.67 × 10^-11) * (2 × 10^30) * (6.3 × 10^4) * (1 / (7.5 × 10^12) - 1 / (4.2 × 10^10)).

7. Calculate the right-hand side of the equation.

8. Take the square root of both sides to solve for v2.

Now, after going through these calculations again, double-check that each step was carried out correctly. Pay attention to the signs and whether the values were substituted accurately. If you still encounter an issue with a negative number under the square root, you may want to check your calculations one more time. Alternatively, you could provide the specific values you obtained for each term in the equation to help identify the error.

It seems like you made a mistake in your calculation. Let's go through the steps again to find out where the error occurred.

First, you correctly set up the equation using the conservation of energy principle:

1/2v1^2 - Gm2/r1 = 1/2v2^2 - Gm2/r2

Next, you rearranged the equation to solve for v2:

1/2v1^2 + Gm2/r1 - Gm2/r2 = 1/2v2^2

To simplify further, you need to combine the terms involving Gm2:

1/2v1^2 + Gm2(1/r1 - 1/r2) = 1/2v2^2

Now, let's substitute the given values into the equation:

v1 = 6.3x10^4 m/s
G = 6.67x10^-11 N(m/kg)^2
m2 (mass of the Sun) = 2x10^30 kg
r2 = 7.5x10^12 m
r1 = 4.2x10^10 m

Plugging these values in, we have:

(1/2)(6.3x10^4)^2 + (6.67x10^-11)(2x10^30)(1/4.2x10^10 - 1/7.5x10^12) = 1/2v2^2

Simplifying this expression gives:

(1/2)(6.3x10^4)^2 + (6.67x10^-11)(2x10^30)(4.2x10^10 - 7.5x10^12) = 1/2v2^2

(1/2)(6.3x10^4)^2 + (6.67x10^-11)(2x10^30)(-7.4578x10^12) = 1/2v2^2

Now, solve for v2 by taking the square root of both sides:

sqrt [(1/2)(6.3x10^4)^2 + (6.67x10^-11)(2x10^30)(-7.4578x10^12)] = v2

By evaluating the right-hand side of this equation using a calculator or computer software, you should obtain the correct value for v2.

Note: It's important to double-check your calculations and make sure all units are consistent throughout the calculations, especially when dealing with powers of 10.