Question

a spaceship has length 120 m, diameter 25 m, and mass4.0 * 103 kg as measured by its crew. as the spaceship moves parallelto its cylindrical axis and passes us, we measure its length to be 90 m.(a) what do we measure its diameter to be? (b) what do we measure themagnitude of its momentum to be?

Answer 1

To calculate the observed length and diameter of the spaceship, we can use the formula for length contraction. The magnitude of the momentum can be calculated using the formula for momentum.

Step-by-step explanation:

The spaceship's length of 120 m as measured by its crew would be shorter when observed by an observer on Earth due to length contraction. To calculate the observed length, we can use the formula for length contraction:

L₂ = L₁ / √(1 - v²/c²)

Where L₂ is the observed length, L₁ is the length as measured by the crew, v is the velocity of the spaceship, and c is the speed of light.

(a) Plugging in the given values, we get:

L₂ = 120 m / √(1 - (0.970c)²/c²)

(b) The magnitude of the spaceship's momentum can be calculated using the formula:

p = mv

Where p is the momentum, m is the mass of the spaceship, and v is its velocity.

Plugging in the given values, we get:

p = (4.0 * 10³ kg) * (0.970c)

Therefore, (a) The observed diameter can be calculated using the formula for length contraction and (b) the magnitude of the momentum can be calculated using the formula for momentum.

Answer 2

The observed diameter of the spaceship is approximately 20.81 meters, and the magnitude of its measured momentum is roughly
`\(1.067 * 10^(12)\)` kg m/s.

(a) Observed Diameter:

Given:

- Rest diameter
`(\( \text{Rest Diameter} \)) = 25 m`

- Observed length
`(\( \text{Observed Length} \)) = 90 m`

- Rest length
`(\( \text{Rest Length} \)) = 120 m`

Using the formula for length contraction:

`\[ v = c * \sqrt{1 - \left(\frac{\text{Observed Length}}{\text{Rest Length}}\right)^2} \]`

First, let's find the velocity v:

`\[ v = 3 * 10^8 \, \text{m/s} * \sqrt{1 - \left((90)/(120)\right)^2} \]`

`\[ v = 3 * 10^8 \, \text{m/s} * \sqrt{1 - \left((3)/(4)\right)^2} \]`

`\[ v = 3 * 10^8 \, \text{m/s} * \sqrt{1 - (9)/(16)} \]`

`\[ v = 3 * 10^8 \, \text{m/s} * \sqrt{(7)/(16)} \]`

`\[ v = 3 * 10^8 \, \text{m/s} * (√(7))/(4) \]`

`\[ v = (3)/(4) * 10^8 \, \text{m/s} * √(7) \]`

`\[ v \approx 1.479 * 10^8 \, \text{m/s} \]`

Now, find the observed diameter
`(\( \text{Observed Diameter} \))` using length contraction:

`\[ \text{Observed Diameter} = \text{Rest Diameter} * \sqrt{1 - (v^2)/(c^2)} \]`

`\[ \text{Observed Diameter} = 25 \, \text{m} * \sqrt{1 - \left(\frac{1.479 * 10^8 \, \text{m/s}}{3 * 10^8 \, \text{m/s}}\right)^2} \]`

`\[ \text{Observed Diameter} = 25 \, \text{m} * √(1 - 0.3074) \]`

`\[ \text{Observed Diameter} = 25 \, \text{m} * √(0.6926) \]`

`\[ \text{Observed Diameter} = 25 \, \text{m} * 0.8324 \]`

`\[ \text{Observed Diameter} \approx 20.81 \, \text{m} \]`

(b) Magnitude of Measured Momentum:

Using the formula for relativistic momentum:

`\[ \text{Momentum} = \frac{m * v}{\sqrt{1 - (v^2)/(c^2)}} \]`

Given:

- Rest mass ( m ) =
`\( 4.0 * 10^3 \) kg`

- Velocity ( v ) ≈
`1.479 * 10^8 \) m/s`

Let's calculate the momentum:

`\[ \text{Momentum} = \frac{(4.0 * 10^3 \, \text{kg}) * (1.479 * 10^8 \, \text{m/s})}{\sqrt{1 - \left(\frac{1.479 * 10^8 \, \text{m/s}}{3 * 10^8 \, \text{m/s}}\right)^2}} \]`

`\[ \text{Momentum} = \frac{5.916 * 10^(11) \, \text{kg m/s}}{√(0.3074)} \]`

`\[ \text{Momentum} \approx \frac{5.916 * 10^(11) \, \text{kg m/s}}{0.5547} \]`

`\[ \text{Momentum} \approx 1.067 * 10^(12) \, \text{kg m/s} \]`

So, the observed diameter of the spaceship would be approximately 20.81 meters, and the magnitude of its measured momentum would be approximately
`\(1.067 * 10^(12)\)` kg m/s.