The average lifetime of a $$ \pi $$ meson in its own frame of reference is 26.0 ns (proper lifetime). If it moves with a speed of 0.95 c ( c is the speed of light) with respect to the Earth (a)what is its lifetime as measured by an observer at rest on Earth? (b)What is the average distance it travels before decaying as measured by an observer at rest on Erath? ( 4 mks ) (a) Lifetime $$ =3.3 \mathrm{~ns} $$, Distance 2.4 m (b) Lifetime $$ =15.5 \mathrm{~ns} $$, Distance 50 m (c) Lifetime $$ =83.3 \mathrm{~ns} $$, Distance 24 m (d) Lifetime $$ =166 . \mathrm{ns} $$, Distance 48 m

Question

UpStudy AI · Accepted Answer

Given:
- The average lifetime of a $$ \pi $$ meson in its own frame of reference is 26.0 ns (proper lifetime).
- The $$ \pi $$ meson moves with a speed of 0.95 c (where c is the speed of light) with respect to the Earth.

We need to find:
(a) The lifetime of the $$ \pi $$ meson as measured by an observer at rest on Earth.
(b) The average distance the $$ \pi $$ meson travels before decaying as measured by an observer at rest on Earth.

To solve this, we can use the concept of time dilation from special relativity. The time dilation formula is given by:

$$ t = \frac{t_0}{\sqrt{1 - \frac{v^2}{c^2}}} $$

where:
- $$ t $$ is the time measured by the observer at rest on Earth.
- $$ t_0 $$ is the proper time (time measured in the $$ \pi $$ meson's own frame of reference).
- $$ v $$ is the speed of the $$ \pi $$ meson with respect to the Earth.
- $$ c $$ is the speed of light.

Let's calculate the lifetime and distance for each part.
The lifetime of the $$ \pi $$ meson as measured by an observer at rest on Earth is approximately 83.3 ns.

Now, let's calculate the average distance the $$ \pi $$ meson travels before decaying as measured by an observer at rest on Earth.
Calculate the value by following steps:
- step0: Calculate:
  $$0.95\times 3\times 10^{8}\times 83.3\times 10^{-9}$$
- step1: Multiply the terms:
  $$237.405\times 10^{8}\times 10^{-9}$$
- step2: Multiply the terms:
  $$237.405\times 10^{8-9}$$
- step3: Subtract the numbers:
  $$237.405\times 10^{-1}$$
- step4: Convert the expressions:
  $$\frac{47481}{200}\times 10^{-1}$$
- step5: Multiply the numbers:
  $$\frac{47481}{2000}$$
The average distance the $$ \pi $$ meson travels before decaying as measured by an observer at rest on Earth is approximately 23.74 m.

Therefore, the correct answer is:
(a) Lifetime = 83.3 ns
(b) Distance = 23.74 m

So, the correct option is (c) Lifetime = 83.3 ns, Distance = 24 m.
(a) Lifetime = 83.3 ns (b) Distance = 24 m

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Physics Questions