Supersonic jets are able to achieve maximum speeds of up to 3.0 × 10-6c. Calculate the percentage contraction in the length of a jet plane travelling at this speed.
To calculate the percentage contraction in the length of a jet plane traveling at a speed close to the speed of light, we can use the relativistic length contraction equation.
The equation for length contraction is given by:
L' = L * sqrt(1 - (v^2/c^2))
Where:
L' is the contracted length
L is the rest length (length at rest)
v is the velocity of the object
c is the speed of light
In this case, the velocity of the jet plane is given as 3.0 × 10^-6c, where c is the speed of light. So we can substitute the values into the equation and calculate the contraction.
L' = L * sqrt(1 - (3.0 × 10^-6c)^2/c^2)
To simplify the equation, we can divide the numerator and denominator of the fraction by c^2:
L' = L * sqrt(1 - (3.0 × 10^-6)^2)
Next, we calculate the square of 3.0 × 10^-6:
(3.0 × 10^-6)^2 = 9.0 × 10^-12
Substituting this value back into the equation:
L' = L * sqrt(1 - 9.0 × 10^-12)
To calculate the contraction percentage, we need to find the difference between the original length L and the contracted length L', and then express that difference as a percentage of the original length.
Percentage contraction = ((L - L') / L) * 100
Substituting the values:
Percentage contraction = ((L - L * sqrt(1 - 9.0 × 10^-12)) / L) * 100
Simplifying this expression further depends on having the specific values for the rest length of the jet plane.
Supersonic jets are able to achieve maximum speeds of up to 3.0 × 10-6c. Calculate the percentage contraction in the length of a jet plane travelling at this speed.
To calculate the percentage contraction in the length of a jet plane traveling at a speed of 3.0 × 10^-6c, we can use the Lorentz contraction formula.
The Lorentz contraction formula is given by:
L' = L * sqrt(1 - (v^2/c^2))
Where:
L' is the contracted length of the object
L is the original length of the object
v is the velocity of the object
c is the speed of light
In this case, we want to find the percentage contraction, which is calculated by:
Percentage Contraction = (L - L') / L * 100
Given:
v = 3.0 × 10^-6c (velocity of the jet)
c = speed of light (3.0 × 10^8 m/s)
Let's substitute the values into the equations:
L' = L * sqrt(1 - (v^2/c^2))
= L * sqrt(1 - (3.0 × 10^-6)^2 / (3.0 × 10^8)^2)
Simplifying,
L' = L * sqrt(1 - 9.0 × 10^-12 / 9.0 × 10^16)
= L * sqrt(1 - 1/10^28)
= L * sqrt(1 - 10^-28)
= L * sqrt(1 - 0.0000000000000000000000000001)
= L * sqrt(0.9999999999999999999999999999)
Since v is extremely small compared to c, we can approximate it as:
L' ≈ L * sqrt(1)
Therefore, the contracted length L' is approximately equal to the original length L.
Now let's calculate the percentage contraction:
Percentage Contraction = (L - L') / L * 100
= (L - L) / L * 100
= 0 / L * 100
= 0%
Hence, the percentage contraction in the length of the jet plane traveling at this speed is approximately 0%.