A balloon rises vertically from a point P on level ground at a uniform speed. After 6 minutes, an observer standing at a point on the ground that is 450√3 metres away from P measures the angle of elevation of the balloon as 60°. Assuming the observation point and P are on the same horizontal level, what is the speed of the balloon in m/s?

Introduction / Context:
This problem illustrates vertical motion combined with an oblique observation from a point on the ground. The balloon rises straight up from its starting point, while the observer stands some horizontal distance away. Using the angle of elevation after a given time, we compute the vertical height and hence the speed of ascent in metres per second.

Given Data / Assumptions:

• The balloon starts from point P on the ground.
• An observer is at a point Q, 450√3 m horizontally from P.
• After 6 minutes, the angle of elevation of the balloon from Q is 60°.
• The balloon rises vertically with uniform speed.
• Ground between P and Q is level.

Concept / Approach:
The balloon's position after 6 minutes forms a right triangle with vertical leg equal to the balloon height and horizontal leg equal to the fixed distance PQ. The angle between PQ and the line of sight is the angle of elevation. Using tan(theta) = opposite / adjacent, we find the height reached in 6 minutes and then divide by the time in seconds to find the speed in m/s.

Step-by-Step Solution:
Horizontal distance PQ = 450√3 m.Angle of elevation after 6 minutes = 60°.Let h be the height of the balloon at that time.Then tan(60°) = h / (450√3).tan(60°) = √3 ⇒ √3 = h / (450√3).So, h = √3 * 450√3 = 450 * 3 = 1350 m.Time taken = 6 minutes = 6 * 60 = 360 seconds.Speed v = distance / time = h / t = 1350 / 360 m/s.Simplify: 1350 / 360 = 135 / 36 = 15 / 4 = 3.75 m/s.

Verification / Alternative check:
We can check tan(60°) from the computed height. Using h = 1350 m and base = 450√3 m, tan(60°) = 1350 / (450√3) = 3 / √3 = √3, as required. This confirms that our triangle is consistent with the given angle.

Why Other Options Are Wrong:
4.25 m/s, 4.5 m/s, 3.45 m/s, 5 m/s: These speeds correspond to different heights over the given 360 seconds and would not yield tan(60°) = √3 when combined with the fixed base of 450√3 m.

Common Pitfalls:
Common mistakes include forgetting to convert minutes to seconds, mixing up which side is opposite and which is adjacent to the angle, or incorrectly simplifying tan(60°). Some learners also mistakenly treat 450√3 as the hypotenuse rather than the base, which leads to incorrect height and speed calculations.

Final Answer:
The speed of the balloon is 3.75 m/s.