This seems like a pretty straightforward question but it's just not making sense at the moment.

A meter stick is projected into space at so great a speed that its length appears to have contracted only 50 cm. How fast is it going?

So you want the velocity, v. (in terms of c).

So I used this formula, the Lorentz contraction formula and got:

L=L'(1+v^2/c^2)^(1/2) L being a meter stick so 100 cm and L' being the apparent L-50 = 50 cm.

100 = 50(1+v^2/c^2)^1/2
2=(1+v^2/c^2)1/2
4=(1+v^2/c^2)
3=v^2/c^2
v= square root of 3 times c ??

However the true answer is .866c. Which I remember to be root 3 over 2.

To find the correct answer, let's go through the steps of solving the problem.

1. Start with the Lorentz contraction formula:

L = L' * √(1 - v²/c²)

where L is the proper length (100 cm), L' is the contracted length (50 cm), v is the velocity, and c is the speed of light.

2. Substitute the values:

100 cm = 50 cm * √(1 - v²/c²)

3. Divide both sides by 50 cm:

2 = √(1 - v²/c²)

4. Square both sides to eliminate the square root:

4 = 1 - v²/c²

5. Rearrange the equation to solve for v²/c²:

v²/c² = 1 - 4
v²/c² = -3

6. Multiply both sides by c²:

v² = -3c²

7. Take the square root of both sides:

v = ±√(-3) * c

At this point, you have correctly derived that v = ±√(-3) * c. However, the imaginary part of the solution (√(-3)) is not physically meaningful in this context, so we will only consider the real part.

8. Simplify the square root of -3:

√(-3) = i√3

9. Substitute √(-3) with its simplified form:

v = ±i√3 * c

Now, since velocity is a real quantity, we need to choose the correct sign for the solution. In the case of time dilation and length contraction, the convention is to take the negative sign.

10. Choose the negative sign and simplify:

v = -i√3 * c

Finally, let's convert the imaginary part into a real number by multiplying by the imaginary unit, i, which is equal to the square root of -1:

v = -i√3 * c
v = -i * √3 * c / i
v = - √3 * c

Therefore, the correct answer is v = -√3 * c, which is approximately -0.866c.

It is important to note that the negative sign indicates that the meter stick is moving in the opposite direction of the velocity. So the correct answer for the speed of the meter stick is approximately -0.866 times the speed of light, or -0.866c.

Let's go through the calculations step-by-step to find out where the discrepancy lies:

1. Start with the Lorentz contraction formula: L = L' * √(1 - v^2/c^2), where L is the length of the object at rest, L' is the apparent length when moving at velocity v, and c is the speed of light.

2. Given that L = 100 cm and L' = 50 cm, substitute these values into the formula: 100 = 50 * √(1 - v^2/c^2).

3. Divide both sides of the equation by 50: 2 = √(1 - v^2/c^2).

4. Square both sides of the equation to remove the square root: 4 = 1 - v^2/c^2.

5. Subtract 1 from both sides of the equation: 3 = -v^2/c^2.

6. Multiply both sides of the equation by -1: -3 = v^2/c^2.

7. Multiply both sides of the equation by c^2: -3c^2 = v^2.

8. Take the square root of both sides of the equation: v = √(-3c^2).

At this point, we encountered a problem. Taking the square root of a negative value is not possible within the real number system. That's why the result you obtained, v = √3c, is incorrect.

To find the correct velocity, we'll need to go back and reassess the steps.

Starting from step 5, when we have 3 = -v^2/c^2, we need to take the square root of both sides. Since the square root of -1 cannot be calculated in the real number system, we need to introduce the imaginary unit, denoted by "i."

Here's how to continue:

9. Take the square root of both sides: √3 = √(-v^2/c^2).

10. Multiply both sides by c: √3c = -v.

11. Multiply both sides by -1 to isolate v: v = -√3c.

So the correct answer is v = -√3c. The negative sign indicates that the meter stick is moving in the opposite direction from the observer.