To solve this problem, we need to determine the initial distance lead created by the thief and then apply the concept of Relative Speed since both the thief and the owner are moving in the same direction.
Step 1: Calculate the Distance Covered by the Thief before Pursuit Starts
The thief drives at a constant speed for half an hour (0.5 hours) before the owner discovers the theft and starts chasing.
Using the distance formula (Distance=Speed×Time):
Initial Lead Distance (D)=40 kmph×0.5 hours=20 km
At the moment the owner begins the chase, the thief is exactly 20 km ahead.
Step 2: Calculate the Relative Speed
Since both individual entities are moving in the same linear direction, the net rate at which the gap is closing is the difference between their speeds:
Relative Speed (vrel)=v2−v1
Relative Speed (vrel)=50 kmph−40 kmph=10 kmph
This means the owner closes the distance gap at a rate of 10 km every hour.
Step 3: Calculate the Time Required to Catch Up
The time required for the owner to cover the initial 20 km lead gap at the relative speed is given by:
Time taken (T)=Relative SpeedInitial Lead Distance
Time taken (T)=10 kmph20 km=2 hours
Final Answer
The owner will catch up with the thief exactly 2 hours after his start.
Therefore, the correct option is (a).
