Which expression is dimensionally consistent with an expression that would yield a value for time \( { }^{-1} \) ? ( \( v \) is velocity, \( x \) is distance, and \( t \) is time) 1) \( v / x \) 2) \( v^{2} / x \) 3) \( x / t \) 4) \( v^{2} t \) 5) \( x / v \)
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To find an expression that yields a value for time \(^{-1}\), let's break down the dimensions of each option. Velocity \(v\) has dimensions of \( [L T^{-1}] \) (length/time), distance \(x\) has dimensions of \( [L] \), and time \(t\) has dimensions of \( [T] \). 1) In \( v/x \), the dimensions are \( [L T^{-1}] / [L] = [T^{-1}] \), which is consistent with time \(^{-1}\) — good choice! 2) In \( v^{2}/x \), the dimensions are \( [L^{2} T^{-2}] / [L] = [L T^{-2}] \)—not what we want. 3) In \( x/t \), the dimensions are \( [L] / [T] = [L T^{-1}] \)—still not \( T^{-1} \). 4) In \( v^{2} t \), the dimensions are \( [L^{2} T^{-2}] [T] = [L^{2} T^{-1}] \)—definitely off. 5) In \( x/v \), the dimensions are \( [L] / [L T^{-1}] = [T] \)—a time value but not \(^{-1}\). So, the dimensionally consistent expression yielding a value for time \(^{-1}\) is \( v/x \).
