A pole of infinite length is planted vertically on the ground. A stone is placed at a distance of 'd1' meters from the pole towards the right. A bird is sitting on the ground exactly in the middle of the pole and stone. The bird flies away towards the right in such a manner that its distance from the pole and the distance between the bird and the stone at any point of time is always equal. Calculate the distance between the bird and it's starting point after n seconds if the bird is covering 'd2'metres horizontally every second.
110 meters
To solve this problem, let's break it down step by step:
1. Start by visualizing the scenario. Draw a vertical line to represent the pole, a point to represent the stone, and another point to represent the initial position of the bird.
2. Since the bird maintains an equal distance from both the pole and the stone, draw a circle centered at the midpoint (midpoint between the pole and the stone). This circle represents the locus of all possible positions of the bird.
3. The bird flies towards the right, covering a horizontal distance of 'd2' meters every second. This means that after 'n' seconds, the bird would have covered a horizontal distance of 'd2 * n' meters.
4. The question asks for the vertical distance between the bird's starting point and its current position after 'n' seconds. Since the bird flies horizontally, it means the vertical distance remains constant.
5. To find the given vertical distance, we need to calculate the height of the triangle formed by the midpoint, the pole, and the bird's starting position. Let's call this height 'h'.
6. We can calculate 'h' using the Pythagorean theorem. Since the triangle is a right-angled triangle, we can use the formula: h^2 + (d1/2)^2 = r^2, where 'r' is the distance between the midpoint and the pole (which is half the distance between the pole and the stone).
7. Solve the above equation to find the value of 'h'.
8. Finally, to find the vertical distance between the bird's starting point and its current position after 'n' seconds, subtract 'h' from the radius of the circle (r) multiplied by (n * d2).
In summary, to find the distance between the bird and its starting point after 'n' seconds, use the formula:
Distance = (r * n * d2) - h
Where:
- 'n' is the number of seconds elapsed.
- 'd2' is the horizontal distance covered by the bird per second.
- 'r' is the radius of the circle (half the distance between the pole and the stone).
- 'h' is the height of the triangle formed by the midpoint, the pole, and the bird's starting position.