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Two cars are traveling on a straight section of dual carriageway in parallel, opposite directions. The perpendicular distance between the cars is a constant 4 meters. Initially, the horizontal distance between the cars is 1km.

Given that each car is traveling at 20ms−1, find the time taken until the actual distance between the cars is first equal to 500m.

Give your answer correct to one decimal place.

Hint

After t seconds have passed, notice that each car has traveled 20t meters. Let the remaining horizontal distance be called x, and the actual distance be â„“:

We want to find the t such that â„“=500.

Hint

Express x in terms of t. Notice that x,â„“ and 4m form a right angled triangle so that

â„“2=x2+42
Solution

After t seconds have passed, notice that each car has traveled 20t meters. Let the remaining horizontal distance be called x, and the actual distance be â„“:

We want to find the t such that â„“=500. Notice that

x=1000−40t

By Pythagoras, we have

x2=ℓ2−16(1000−40t)2=5002−161000−40t=±24998440t=1000±24434t=25±3569

We want the first time ℓ=50, so we need the smaller of the two solution, which is 12.5(1 d.p.)