(II) At what speed v will the length of a 1.00-m stick look 10.0% shorter (90.0 cm)?

Verified step by step guidance

1. Identify the concept involved: This problem is based on the concept of length contraction in special relativity, which states that the length of an object moving at a significant fraction of the speed of light will appear contracted to a stationary observer.
2. Set up the length contraction formula: The formula to calculate the contracted length (L') is L' = L \sqrt{1 - \frac{v^2}{c^2}}, where L is the proper length (original length), v is the velocity of the object, and c is the speed of light.
3. Insert the given values into the formula: In this case, the proper length L is 1.00 m, and the contracted length L' is 0.90 m (90.0 cm).
4. Rearrange the formula to solve for v: Isolate v on one side of the equation to find v = c \sqrt{1 - \left(\frac{L'}{L}\right)^2}.
5. Substitute the values of L and L' into the rearranged formula and solve for v: This will give you the speed at which the stick appears 10% shorter.

Key Concepts

Length Contraction

Length contraction is a phenomenon predicted by Einstein's theory of special relativity, where an object in motion is measured to be shorter along the direction of motion compared to its length at rest. This effect becomes significant as the object's speed approaches the speed of light, leading to observable differences in length for fast-moving objects.

Lorentz Factor

The Lorentz factor, denoted as γ (gamma), is a crucial component in special relativity that quantifies the amount of time dilation and length contraction experienced by an object moving at a significant fraction of the speed of light. It is calculated using the formula γ = 1 / √(1 - v²/c²), where v is the object's speed and c is the speed of light.

Relative Velocity

Relative velocity refers to the velocity of one object as observed from another object. In the context of special relativity, it is essential to understand how the speed of an object affects its observed properties, such as length and time, depending on the frame of reference of the observer.