A football has a diameter of 22cm. If the ball is travelling at a speed of 0.987c relative to the Earth, what is the length of it's smallest dimension as measured on the Earth? Use cm as the unit and give units.
Question
A football has a diameter of 22cm. If the ball is travelling at a speed of 0.987c relative to the Earth, what is the length of it's smallest dimension as measured on the Earth? Use cm as the unit and give units.
Solution
The length contraction formula is given by:
L = L0 * sqrt(1 - v^2/c^2)
where: L is the contracted length, L0 is the rest length (or the original length), v is the velocity of the object, c is the speed of light.
Given: L0 = 22 cm (the diameter of the football), v = 0.987c (the speed of the football relative to the Earth), c = 1 (when we use c as the unit for v, c becomes 1 in these units).
Substituting these values into the formula, we get:
L = 22 cm * sqrt(1 - (0.987)^2) L = 22 cm * sqrt(1 - 0.974169) L = 22 cm * sqrt(0.025831) L = 22 cm * 0.16072
Therefore, the length of the football's smallest dimension as measured on the Earth is approximately 3.54 cm.
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