Special relativity and length contraction

[wumple]

Homework Statement

Consider a universe in which the speed of light c is equal to 100 mi/h. A lincoln continental traveling at a speed v relative to a fixed radar speed trap overtakes a Volkswagen traveling at the speed limit of 50 mi/h. The lincoln's speed is such that its length, as measured by the fixed observer, is the same as that of the Volkswagen. By how much is the Lincoln exceeding the speed limit? At rest, a Lincoln is twice as long as a Volkswagen.

Homework Equations

Length contraction L = gamma Lo

The Attempt at a Solution

I set L = .5 Lo which makes gamma = 2, solving that gives me v = .866c and that the Lincoln is speeding by 36.6 mi/h. The answer in the back of the book is 40 mi/h. Did I do something wrong, or is it just a significant figure difference?

 

[anathema]

Thank you for your interesting question. I would like to provide you with a thorough explanation of the solution to this problem.

Firstly, let's review the given information. We are in a universe where the speed of light is 100 mi/h, and we have a Lincoln Continental traveling at a speed v relative to a fixed radar speed trap. The Lincoln overtakes a Volkswagen traveling at the speed limit of 50 mi/h. The length of the Lincoln, as measured by the fixed observer, is the same as that of the Volkswagen. We are asked to find how much the Lincoln is exceeding the speed limit.

To solve this problem, we will use the equation for length contraction, L = gamma * Lo, where L is the length observed by the fixed observer, gamma is the Lorentz factor, and Lo is the length of the object at rest.

We are given that at rest, a Lincoln is twice as long as a Volkswagen. Therefore, Lo for the Lincoln is twice the Lo for the Volkswagen.

Now, we can use the given information to find the value of gamma. Since the Lincoln is traveling at a speed of v, we can use the equation v = c * gamma, where c is the speed of light. Substituting the given values, we have v = 100 mi/h * gamma.

We also know that the Lincoln overtakes the Volkswagen, which means that it has traveled the same distance as the Volkswagen in a shorter amount of time. This can be expressed as v * t = 50 mi, where t is the time taken by the Lincoln to overtake the Volkswagen.

Substituting the value of v, we get 100 mi/h * gamma * t = 50 mi.

Solving for t, we get t = 0.5/gamma.

To find the value of gamma, we can use the equation v = c * gamma again. Substituting the given values, we have v = 100 mi/h * gamma = 50 mi, which gives us gamma = 0.5.

Now, we can use the equation for length contraction to find the value of L for the Lincoln. Substituting the values of gamma and Lo, we have L = 0.5 * 2Lo = Lo.

Since the length of the Lincoln is the same as the length of the Volkswagen, we can equate the two lengths, which gives us Lo = 50