Question

Which of the following sets of displacements might be capable of bringing a car to its starting point?
(A) ${\text{4,6,8 and 15 km}}$
(B) ${\text{10,30,50 and 120 km}}$
(C) ${\text{5,10,30 and 50 km}}$
(D) ${\text{40,50,75 and 200 km}}$

Answer 1

Hint: A car can return to its starting point, only when the sum of three displacements is greater than the value of the maximum displacement. In any other case, this would not be possible.

Complete Step by Step Solution: It has been given that a set of displacements has been arranged such that a car would be able to come back to its starting point.
For the second set, we have ${\text{10,30,50 and 120 km}}$. The maximum value of displacement is 120km. Even if the car travels in a straight line, the sum of the first three displacements shall be,
$10 + 30 + 50 = 90km$.
The sum of the three displacements is less than the value of the maximum displacement. $90km < 120km$.
Thus, it cannot be possible.
For the third set, we have ${\text{5,10,30 and 50 km}}$. The maximum value of displacement is 50km. Even if the car travels in a straight line, the sum of the first three displacements shall be,
$5 + 10 + 30 = 45km$.
The sum of the three displacements is less than the value of the maximum displacement. $45km < 50km$.
Thus, it cannot be possible.
For the fourth set, we have ${\text{40,50,75 and 200 km}}$. The maximum value of displacement is 200km. Even if the car travels in a straight line, the sum of the first three displacements shall be,
$40 + 50 + 75 = 165km$.
The sum of the three displacements is less than the value of the maximum displacement. $165km < 200km$.
Thus, it cannot be possible.
For the first set of displacements, we have ${\text{4,6,8 and 15 km}}$.The maximum value of displacement is 15km. Even if the car travels in a straight line, the sum of the first three displacements shall be,
$4 + 6 + 8 = 18km$.
The sum of the three displacements is more than the value of the maximum displacement. $18km > 15km$.
Thus, this is the only case where this set of displacements might be capable of bringing a car to its starting point.

Hence the correct answer is Option A.

Note: If an object moves relative to a reference frame—for example, if a professor moves to the right relative to a whiteboard, or a passenger moves toward the rear of an airplane—then the object’s position changes. This change in position is known as displacement. The word displacement implies that an object has moved, or has been displaced. Displacement is defined to be the change in position of an object. Displacement is a vector. This means it has a direction as well as a magnitude and is represented visually as an arrow that points from the initial position to the final position.