An unstable particle is created in the upper atmosphere from a cosmic ray and travels straight down toward the surface of the earth with a speed of 0.99540c relative to the earth. A scientist at rest on the earth’s surface measures that the particle is created at an altitude of 45.0 km. (a) As measured by the scientist, how much time does it take the particle to travel the 45.0 km to the surface of the earth? (b) Use the length-contraction formula to calculate the distance from where the particle is created to the surface of the earth as measured in the particle’s frame. (c) In the particle’s frame, how much time does it take the particle to travel from where it is created to the surface of the earth? Calculate this time both by the time dilation formula and from the distance calculated in part (b). Do the two results agree?

Verified step by step guidance

1. Step 1: For part (a), calculate the time it takes for the particle to travel 45.0 km as measured by the scientist on Earth. Use the formula for time: $t=\frac{d}{v}$, where $d$ is the distance (45.0 km) and $v$ is the speed of the particle (0.99540c). Convert the distance to meters and use the speed of light $c$ = 3.00 × 10⁸ m/s for calculations.
2. Step 2: For part (b), use the length-contraction formula to calculate the distance as measured in the particle’s frame. The formula is $L_p=L_e\sqrt{1-{\frac{v}{c}}^2}$, where $L_e$ is the distance measured by the scientist (45.0 km), $v$ is the speed of the particle, and $c$ is the speed of light.
3. Step 3: For part (c), calculate the time in the particle’s frame using the time dilation formula. The formula is $t_p=\frac{t_e}{\sqrt{1-{\frac{v}{c}}^2}}$, where $t_e$ is the time measured by the scientist in part (a).
4. Step 4: Alternatively, calculate the time in the particle’s frame using the contracted distance from part (b) and the particle’s speed. Use the formula $t_p=\frac{L_p}{v}$, where $L_p$ is the contracted distance and $v$ is the speed of the particle.
5. Step 5: Compare the two results for the time in the particle’s frame from step 3 and step 4. Verify that they agree, demonstrating consistency between the time dilation formula and the contracted distance approach.

Key Concepts

Relativity of Time and Length

In the theory of relativity, time and space are not absolute but relative to the observer's frame of reference. This means that time can appear to pass at different rates for observers in different states of motion, and lengths can contract depending on the relative velocity between observers. This is crucial for understanding how measurements differ between the scientist on Earth and the particle moving at a significant fraction of the speed of light.

Time Dilation

Time dilation is a phenomenon predicted by Einstein's theory of relativity, where a clock moving relative to an observer ticks slower than a clock at rest with respect to that observer. For the particle traveling at 0.99540c, the time experienced by the particle will be less than the time measured by the scientist on Earth. This concept is essential for calculating the time it takes for the particle to reach the Earth's surface from both frames of reference.

Length Contraction

Length contraction is another relativistic effect where the length of an object in motion is measured to be shorter than its length at rest, as observed from a stationary frame. The formula for length contraction is L = L0√(1 - v²/c²), where L0 is the proper length, v is the relative velocity, and c is the speed of light. This concept is necessary for determining the distance the particle travels as measured in its own frame of reference.