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 solve this problem, we can use the formula for relativistic momentum:
p = γmv
Where:
p = relativistic momentum
γ = Lorentz factor
m = mass of the object
v = velocity of the object
Here, the observer measures the relativistic momentum of the woman to be 2.2 × 10^10 kg·m/s.
Since the woman is moving past the observer in the direction parallel to her height, we can assume her velocity is close to the speed of light, which is approximately 3 × 10^8 m/s.
Now, we can rearrange the formula and solve for the Lorentz factor (γ):
γ = p / (mv)
γ = (2.2 × 10^10 kg·m/s) / (47 kg * 3 × 10^8 m/s)
γ ≈ 15.02
Next, we can use the Lorentz factor to find the contracted height of the woman as observed by the observer. The formula for length contraction is given by:
L = L0 / γ
Where:
L = contracted length
L0 = proper length (original length)
In this case, the contracted length is the height of the woman as measured by the observer. Let's assume her proper height (original length) is H.
H = L / γ
Now we can substitute the given values:
H = 1.6 m / 15.02
H ≈ 0.1064 m
Hence, the observer would measure the height of the woman to be approximately 0.1064 meters.
To determine the height of the woman measured by the observer, we need to consider the relativistic momentum and the velocity of the woman.
The relativistic momentum (p) is defined as the product of the rest mass (m) and the velocity (v) of an object. It can be calculated using the equation:
p = γm0v,
where γ is the Lorentz factor, m0 is the rest mass, and c is the speed of light.
Since the observer measures the relativistic momentum of the woman to be 2.2 × 10^10 kg·m/s, we can use this information to solve for the velocity (v).
First, we need to calculate the rest mass (m0) using the mass (m) of the woman. The rest mass is the mass of an object when it is at rest, and it can be calculated using the equation:
m0 = γm
We can rearrange this equation to solve for γ:
γ = m0/m.
Using the given values, we have:
m0 = 47 kg
m = 1.6 m
Substituting these values into the equation, we get:
γ = 47 kg / 1.6 m = 29.375
Now, we can calculate the velocity (v) using the equation for relativistic momentum:
p = γm0v
Rearranging this equation to solve for v, we get:
v = p / (γm0).
Substituting the given values, we have:
p = 2.2 × 10^10 kg·m/s
γ = 29.375
m0 = 47 kg
Now we can calculate v:
v = (2.2 × 10^10 kg·m/s) / (29.375 * 47 kg) ≈ 1.266 × 10^7 m/s.
Finally, to find the height of the woman as measured by the observer, we multiply the velocity (v) by the time (t) during which the observer measures the height. Since the movement is parallel to her height, the time the observer measures is equal to her height divided by the velocity:
t = 1.6 m / (1.266 × 10^7 m/s) ≈ 1.26 × 10^-7 seconds.
Therefore, the observer measures the height of the woman to be approximately 1.26 × 10^-7 seconds.