If a clock in an airplane is found to slow down by 5 parts in 1013, (i.e., ∆t/∆to = 1.0 + 5.0 × 10. −13), at what speed is the airplane travelling? (Hint: You might need to use an expansion for y)
To solve this problem, we can use the equation for time dilation, which relates the time experienced by an observer in relative motion to the speed of the object. The equation is:
∆t/∆to = √(1 - v^2/c^2)
Where:
∆t is the time observed in the moving frame,
∆to is the time observed in the stationary frame,
v is the velocity of the object (airplane), and
c is the speed of light.
Given that ∆t/∆to = 1.0 + 5.0 × 10^−13, we can rewrite the equation as:
(1.0 + 5.0 × 10^−13) = √(1 - v^2/c^2)
To find the velocity (v), we need to rearrange the equation and solve for v. Let's walk through the steps:
Start by squaring both sides of the equation:
(1.0 + 5.0 × 10^−13)^2 = 1 - v^2/c^2
Expand the left side of the equation using the binomial expansion:
1 + 2(5.0 × 10^−13) + (5.0 × 10^−13)^2 = 1 - v^2/c^2
Simplify the expansion:
1 + 1.0 × 10^−12 + 2.5 × 10^−25 = 1 - v^2/c^2
Rearrange the equation to isolate v^2/c^2 on one side:
v^2/c^2 = 1 - (1 + 1.0 × 10^−12 + 2.5 × 10^−25)
Simplify the right side of the equation:
v^2/c^2 = -1.0 × 10^−12 - 2.5 × 10^−25
Since the velocity of an object cannot be negative, we can discard the negative sign:
v^2/c^2 = 1.0 × 10^−12 + 2.5 × 10^−25
Now, we can solve for v by taking the square root of both sides:
v/c = √(1.0 × 10^−12 + 2.5 × 10^−25)
To calculate the actual value of v, we need to know the value of c, which is the speed of light. The speed of light in vacuum is approximately 3.0 × 10^8 meters per second.
So, substituting the known values, we get:
v/3.0 × 10^8 = √(1.0 × 10^−12 + 2.5 × 10^−25)
Solving for v:
v = 3.0 × 10^8 × √(1.0 × 10^−12 + 2.5 × 10^−25)
Finally, calculate the value of v using a calculator or any appropriate software.