Multiple choice

A man running with a uniform speed 'u' on a straight road observes a stationary bus at a distance 'd' ahead of him. At that instance , the bus starts with an acceleration 'a' .The condition that he would be able to catch the bus is:

  1. $\displaystyle d \, \leq \, \frac{u^2}{a}$
  2. $\displaystyle d \, \leq \, \frac{u^2}{2a}$
  3. $\displaystyle d \, \leq \, \frac{u^2}{3a}$
  4. $\displaystyle d \, \leq \, \frac{u^2}{4a}$
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B Correct answer
Explanation

The man catches the bus if his distance covered in time t is greater than or equal to the bus's distance plus the initial gap d. ut >= 0.5at^2 + d. Rearranging: 0.5at^2 - ut + d <= 0. For a real solution for t, the discriminant must be >= 0. (-u)^2 - 4(0.5a)(d) >= 0, so u^2 - 2ad >= 0, which means d <= u^2 / 2a.

AI explanation

Using the equation of motion, the distance covered by the man is ut while the bus covers 0.5at squared. For the man to catch the bus, the equation ut equals d plus 0.5at squared must have a real, positive solution for time t. Rearranging this into a standard quadratic form gives 0.5at squared minus ut plus d equals 0, which requires the discriminant (u squared minus 2ad) to be greater than or equal to zero. Solving this inequality yields the condition that d must be less than or equal to u squared divided by 2a. The result is d <= u^2 / 2a.

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