A research team operates an aerial drone and a ground rover along a straight track of length L meters between Base 1 an…

To find the distance d from Base 1 where the drone and rover meet, express the total time elapsed from t = 0 until the meeting point, T, for both vehicles and set them equal.

1. Rover's timeline:

The rover departs Base 1 at t = c and travels a distance d at speed v. The travel time is d / v, so the total time at the moment they meet is:

T = c + d / v

2. Drone's timeline:

- Time to reach Base 2: The drone covers distance L at speed 2v, taking L / (2v) seconds.

- Hover delay at Base 2: b seconds.

- Return flight to meeting point: The meeting point is d meters from Base 1, so the drone travels a return distance of L - d meters from Base 2 toward Base 1 at speed v. The return travel time is (L - d) / v.

Therefore, the total time at the moment they meet is:

T = L / (2v) + b + (L - d) / v

3. Equating the two expressions for T:

c + d / v = L / (2v) + b + (L - d) / v

Subtract c from both sides:

d / v = L / (2v) + (b - c) + (L - d) / v

Multiply the entire equation by 2v to clear denominators:

2d = L + 2v(b - c) + 2(L - d)

2d = L + 2v(b - c) + 2L - 2d

2d = 3L + 2v(b - c) - 2d

Add 2d to both sides:

4d = 3L + 2v(b - c)

d = (3L + 2v(b - c)) / 4

Choice A is incorrect because it incorrectly assumes the drone flies at speed 2v for both legs of its flight, setting the drone's travel time to (2L - d) / (2v) + b, which leads to 3d = 2L + 2v(b - c).

Choice C is incorrect because it reverses the signs of the two time delays, producing (c - b) instead of (b - c).

Choice D is incorrect because it adds the rover's departure delay c to the drone's turnaround delay b rather than subtracting it.