To determine the value of v for which γ = 1.010, we can use the equation for time dilation:
γ = 1 / sqrt(1 - (v^2 / c^2))
Here, γ represents the Lorentz factor, v is the velocity of the object, and c is the speed of light in a vacuum, which is approximately 299,792,458 meters per second.
To find the value of v, we can rearrange the equation as follows:
1.010 = 1 / sqrt(1 - (v^2 / c^2))
Squaring both sides of the equation, we get:
1.0201 = 1 / (1 - (v^2 / c^2))
Now, let's solve for v. Multiply both sides of the equation by (1 - (v^2 / c^2)):
1.0201 - 1.0201(v^2 / c^2) = 1
Rearrange the equation:
1.0201(v^2 / c^2) = 0.0201
Divide both sides of the equation by 1.0201:
(v^2 / c^2) = 0.0201 / 1.0201
(v^2 / c^2) = 0.0197
Multiply both sides of the equation by c^2:
v^2 = 0.0197 * c^2
Now, take the square root of both sides of the equation to isolate v:
v = sqrt(0.0197 * c^2)
v ≈ sqrt(0.0197) * c
v ≈ 0.1405 * c
Finally, substitute the value of c to obtain the approximate value of v:
v ≈ 0.1405 * 299,792,458 m/s
v ≈ 42,157,734.79 m/s
Therefore, for a speed of approximately 42,157,734.79 meters per second, the Lorentz factor γ will be equal to 1.010, resulting in time dilation and length contraction effects amounting to less than 1%.