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Chapter 36: Q. 25 (page 1060)

A human hair is about $50 渭 m$ in diameter. At what speed, in m/s, would a meter stick 鈥渟hrink by a hair鈥�?
Hint: Use the binomial approximation.

Short Answer

Expert verified

At a speed of $3 脳 10^{6} m / s$ , a meter stick shrink by a hair.

Step by step solution

01

Given Information

We have given that a human hair is about $50 渭 m$ in diameter.

We must determine at what pace in m/s a meter stick "shrinks by a hair" using the binomial approximation.

02

Simplify

$S$ : Grounds reference frame

$S'$ : The meter establishes a frame of reference.

$L' = l (inframe S') .$

On the ground, an experimenter measures the length to be constricted to:

$L = \sqrt{1 - 尾^{2} l} 鈮� (1 - \frac{1}{2} 尾^{2}) l$

$l - L = \frac{1}{2} 尾^{2} l$

As a result, $尾$ has the following value:

$尾 = \sqrt{\frac{2 (l - L)}{l}} 尾 = \sqrt{\frac{2 (50 脳 10^{- 6} m)}{1.00 m}} 尾 = 0.01$

Now, we must determine the value of $v$ using the following relation:

$v = 尾路 c v = 0.01 脳 (3 脳 10^{8} m / s) v = 3 脳 10^{6} m / s$

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Most popular questions from this chapter

Consider the inelastic collision $e^{-} + e^{-} 鈫� e^{-} + e^{-} + e^{-} + e^{+}$ in which an electron-positron pair is produced in a head-on collision between two electrons moving in opposite directions at the same speed. This is similar to Figure 36.39, but both of the initial electrons are moving.

a. What is the threshold kinetic energy? That is, what minimum kinetic energy must each electron have to allow this process to occur?

b. What is the speed of an electron with this kinetic energy?

You are flying your personal rocket craft at $0.90 c$ from Star $A$ toward Star $B .$ The distance between the stars, in the stars鈥� reference frame, is $1.0 l y .$ Both stars happen to explode simultaneously in your reference frame at the instant you are exactly halfway between them. Do you see the flashes simultaneously? If not, which do you see first, and what is the time difference between the two?

Many science fiction spaceships are powered by antimatter reactors. Suppose a $20 m$ long spaceship, with a mass of $15000 kg$

when empty, carries $2000 kg$ of fuel: $1000 kg$ each of matter and antimatter. The matter and antimatter are slowly combined, and the energy of their total annihilation is used to propel the ship. After consuming all the fuel and reaching top speed, the spaceship flies past a space station that is stationary with respect to the planet from which the ship was launched. What is the length of the spaceship as measured by astronauts on the space station?

At what speed, as a fraction of c, must an electron move so that its total energy is 10% more than its rest mass energy?

A rocket travels in the $x$ -direction at speed $0.60 c$ with respect to the earth. An experimenter on the rocket observes a collision between two comets and determines that the spacetime coordinates of the collision are $(x^{'}, t^{'}) = (3.0 脳 10^{10} m, 200 s)$ . What are the spacetime coordinates of the collision in earth鈥檚 reference frame?

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