Q) A hot air balloon is rising vertically from a point A on the ground which is at distance of 100m from a car parked at a point P on the ground. Amar, who is riding the balloon, observes that it took him 15 seconds to reach a point B which he estimated to be equal to the horizontal distance of his starting point from the car parked at P.
i. Find the angle of depression from the balloon at a point B to the car at point P.
ii. Find the speed of the balloon?
ii. Find the total time taken by the balloon to reach the point C from ground?
iii. After certain time Amar observes that the angle of depression is 60. Find the vertical distance travelled by the balloon during this time
(i) Angle of Depression to point P from Point B:
Angle of depression = ∠BPA
In Δ BAP, tan θ = = 1
Since we know that tan 45° = 1,
∴ θ = 45°
Therefore, Angle of Depression to point P from Point B is 45°.
ii. Calculate the speed of the balloon:
Since we know that the speed =
Therefore, the speed = = 6.67 m/sec
Therefore, the speed of the balloon = 6.67 m/sec
ii. Total time taken by the balloon to reach the point C from ground:
To get time, let’s first calculate distance CA travelled by balloon:
In Δ CAP, tan 60 =
∴ √3 =
∴ CA = 100 √3 m
Now, since it is given that balloon travelled 100 m in 15 sec (from ground to point B),
∴ it will travel 100 √3 m in = 15 √3
Therefore, total time taken by the balloon to reach the point C from ground will be 15 √3 seconds
(iii) Vertical distance travelled by the balloon during this time:
After certain time means observer’s location is changing from point B to point C and we need to calculate distance BC.
We just calculated in part (ii) 2nd option, that distance AC = 100√3 m
Therefore distance BC = AC – AB = 100√3 – 100 = 100 (√3 -1) m