v = 0.98c → v²/c² = 0.9604. Then 1 − v²/c² = 1 − 0.9604 = 0.0396. √0.0396 ≈ 0.199. γ = 1 / 0.199 ≈ 5.03.

["Understanding Time Dilation: Deriving γ from v = 0.98c and Relativistic Effects", "When objects move at significant fractions of the speed of light, relativistic effects become prominent—none more critical than time dilation and length contraction. A common calculation in special relativity involves the Lorentz factor, γ (gamma), defined as:", "[\n\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\n]", "where ( v ) is the object’s velocity and ( c ) is the speed of light.", "### From ( v = 0.98c ) to Relativistic Ratios", "Let’s begin by substituting ( v = 0.98c ) into the Lorentz factor formula:", "[\n\frac{v^2}{c^2} = (0.98)^2 = 0.9604\n]", "Now compute the relativistic velocity squared ratio:", "[\n1 - \frac{v^2}{c^2} = 1 - 0.9604 = 0.0396\n]", "Next, take the square root of this quantity to find the inverse Lorentz factor:", "[\n\sqrt{1 - \frac{v^2}{c^2}} = \sqrt{0.0396} \approx 0.199\n]", "Thus, the Lorentz factor γ is:", "[\n\gamma = \frac{1}{0.199} \approx 5.03\n]", "### What This Means in Physics", "The value ( \gamma \approx 5.03 ) indicates how strongly time is dilated at ( 0.98c ) relative to a stationary observer. From this γ, we can see that time slows down significantly: clocks moving at 98% of light speed tick about 5 times slower compared to the observer’s frame.", "For example, one second for the moving object equals approximately 5.03 seconds for the observer. This is a core prediction of Einstein’s special relativity and has been confirmed in particle accelerators and satellite-based experiments.", "### Why This Matters", "Understanding that ( v = 0.98c ) leads to ( \frac{v^2}{c^2} = 0.9604 ) and ( \gamma \approx 5.03 ) enables students, researchers, and anyone interested in modern physics to grasp the profound non-intuitive behavior of time and space at relativistic speeds.", "Knowing how to derive and interpret these values is essential for both theoretical understanding and practical applications involving high-speed travel, GPS satellite corrections, and cosmic phenomena.", "---", "Summary:\nWhen an object moves at 0.98 times the speed of light,\n[\n\frac{v^2}{c^2} = 0.9604\n]\nand the Lorentz factor γ is approximately 5.03, reflecting dramatic time dilation. This demonstrates one of relativity’s most striking predictions."]