A woman is 1.6 m tall and has a mass of 47 kg. She moves past an observer with the direction of the motion parallel to her height. The observer measures her relativistic momentum to have a magnitude of 2.2 × 1010 kg·m/s. What does the observer measure for her height?
To find the height observed by the observer, we need to use the relativistic momentum equation:
p = γm0v
Where:
p is the relativistic momentum observed by the observer
γ is the Lorentz factor
m0 is the rest mass of the object
v is the velocity of the object
Given:
p = 2.2 × 10^10 kg·m/s
m0 = 47 kg
Since the motion is parallel to her height, we can assume that the velocity is the speed of light, c.
Using the equation for relativistic momentum, we can rearrange it to find γ:
γ = p / (m0v)
Substituting the given values:
γ = (2.2 × 10^10 kg·m/s) / (47 kg × c)
To solve this, we need to calculate the value of c, which is the speed of light. The speed of light is approximately 3 × 10^8 m/s.
Substituting the value of c:
γ = (2.2 × 10^10 kg·m/s) / (47 kg × (3 × 10^8 m/s))
Simplifying:
γ ≈ 1.47615 × 10^2
Now, we can use the Lorentz contraction equation to find the observed height, h:
h = h0 / γ
Where:
h is the observed height
h0 is the rest height of the object
Given:
h0 = 1.6 m
Substituting the given values:
h = (1.6 m) / (1.47615 × 10^2)
To find the observed height, we can solve this equation:
h ≈ 1.08328 × 10^-2 m
Therefore, the observer measures her height to be approximately 1.08328 × 10^-2 meters.
To find the height as measured by the observer, we need to use the concept of relativistic momentum and Lorentz transformation equations. First, let's calculate the woman's initial momentum.
The formula for momentum in special relativity is:
p = γ * m * v
Where:
p is the momentum
m is the mass
v is the velocity
γ is the Lorentz factor and is given by γ = 1/√(1 - v²/c²), where c is the speed of light.
As the observer measures the momentum to be 2.2 × 10^10 kg·m/s, we can equate this with the relativistic momentum formula:
2.2 × 10^10 kg·m/s = γ * m * v
Next, let's solve for the velocity:
v = (2.2 × 10^10 kg·m/s) / (γ * m)
To calculate γ and get an accurate value for v, we need to determine v using the relativistic momentum formula:
p = γ * m * v
As we know p and m, we can solve for γ * v:
γ * v = p / m = (2.2 × 10^10 kg·m/s) / (47 kg)
Now, we can use the following formula to calculate the Lorentz factor:
γ = 1 / √(1 - v²/c²)
Rearranging the formula, we get:
1 - v²/c² = 1 / γ²
Since c, the speed of light, is a known constant (approximately 299,792,458 m/s), we can substitute these values into the equation:
1 - (v / 299,792,458 m/s)² = 1 / γ²
Now, let's solve for γ:
γ² = 1 / (1 - (v / 299,792,458 m/s)²)
γ = √(1 / (1 - (v / 299,792,458 m/s)²))
By substituting the value of v we found earlier into the equation, we can find γ.
Finally, we can use the Lorentz transformation equation to calculate the height as measured by the observer:
h' = h * γ
Where:
h' is the observed height
h is the actual height
By substituting the actual height (1.6 m) and the calculated γ value into the equation, we can find the answer.