Answer:
Let the passenger catches the bus after time t. Distance travelled by the bus in time t, \[{{s}_{1}}=0+\frac{1}{2}a{{t}^{2}}=\frac{1}{2}a{{t}^{2}}\] ... (i) Distance travelled by the passenger, \[{{s}_{2}}=\upsilon t+0=\upsilon t\] ...(ii) The passenger will catch the bus if, \[d={{s}_{1}}={{s}_{2}}\] or \[d+\frac{1}{2}a{{t}^{2}}=\upsilon t\] or \[\frac{1}{2}a{{t}^{2}}-\upsilon t+d=0\] or \[t=\frac{\left[ \upsilon \pm \sqrt{{{\upsilon }^{2}}-2ad} \right]}{a}\]. The passenger will catch the bus if t is real , i.e., \[{{\upsilon }^{2}}\ge 2ad\] or \[\upsilon \ge \sqrt{2ad}\] Thus, minimum speed of passenger for catching the bus is \[\sqrt{2ad}\].
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