An eagle is sitting at the top of a 100 m high vertical cliff and a mouse is at the base of the cliff. The mouse starts running away from the cliff on a level ground in a straight line at a speed 10 m/s. The eagle spots the mouse and dives at 45∘ to the ground at a speed of 20 m/s and captures the mouse. The time at which the eagle started the dive is

Correct option is A

Solution:

Step 1: Key observations

• Height of the cliff: 100 m.
• Speed of the mouse: 10 m/s.
• Speed of the eagle: 20 m/s, diving at an angle of 45° to the ground.
• The eagle's velocity can be broken into two components:
  • Horizontal velocity = 20 / √2 ≈ 14.14 m/s.
  • Vertical velocity = 20 / √2 ≈ 14.14 m/s.

Step 2: Time for the eagle to descend

The eagle takes time to cover the vertical height of 100 m. Time is calculated as:

Time = height / vertical velocity = 100 / 14.14 ≈ 7.07 seconds.

Step 3: Distance traveled by the mouse in this time

The mouse runs on level ground at a speed of 10 m/s. In 7.07 seconds, the mouse travels:

Distance = speed × time = 10 × 7.07 ≈ 70.7 meters.

Step 4: Matching the eagle's horizontal distance

The eagle must also cover the horizontal distance traveled by the mouse.

The horizontal distance covered by the eagle in 7.07 seconds is:

Distance = horizontal velocity × time = 14.14 × 7.07 ≈ 100 meters.

Step 5: Delay in the eagle's dive

If the eagle starts its dive immediately, it will overshoot the mouse. The eagle needs to delay its dive.

The required delay time is calculated as:

Delay = (distance difference) / mouse's speed.Delay = (100 - 70.7) / 10 ≈ 3 seconds.

Final Answer:

The eagle starts its dive approximately 3 seconds after the mouse starts running.

Answer: (a)