An object begins moving with speed v0 toward a destination that is a distance D

Question

An object begins moving with speed v0 toward a destination that is a distance D from its starting point. The speed of the object decreases in proportion to how far it has traveled until, upon arrival at the destination, its speed is zero. If x represents how far the object has traveled from the starting point, then the speed of the object v depends of x according to the following relation,
v = V0(1-X/D)
Note that, as described above, when x = 0 this evaluates to v = v0 and when the object reaches the destination, x = D this evaluates to v = 0. Determine at what time the object will be half way to the destination if D = 50 mi and v0 = 50 mph. (Answer 0.69 hr)

Explanation / Answer

v = V0(1-X/D)

For X= 0.5D

v = V0(1-0.5D/D) = 0.5V0

Vave = ( v + V0)/2 = (0.5V0 + V0)/2 = (0.5*50+ 50)/2 = 37.5 mph

t = X/Vave = 0.5D/Vave = 25mi/37.5mph = 0.67hr

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