To solve the problem of when the lion catches the deer, we need to analyze their movements mathematically. The deer starts at the origin (0,0) and runs vertically along the y-axis at a speed of y m/s. The lion starts at (100,0) and chases the deer at a speed of x m/s, where x is greater than y. The lion's path will change as it continuously adjusts its direction to run towards the deer. Let's break this down step by step.
Understanding the Movement
The deer moves straight up the y-axis, while the lion moves diagonally towards the deer. Initially, the lion is at (100,0) and the deer is at (0,0). As time progresses, the positions of both animals change:
- The position of the deer at time t is given by (0, yt).
- The position of the lion can be represented as (100 - xt, 0) initially, but as it moves towards the deer, its coordinates will change based on its angle of pursuit.
Setting Up the Equations
At any time t, the lion's position can be described using the angle θ that it makes with the x-axis. The lion's speed can be broken down into its x and y components:
- The x-component of the lion's speed is x * cos(θ).
- The y-component of the lion's speed is x * sin(θ).
As the lion chases the deer, the angle θ changes. Initially, θ is 180 degrees (pointing directly left), and as the lion moves, θ approaches 90 degrees (pointing directly up). The lion's position at time t can be expressed as:
- x-coordinate: 100 - x * t * cos(θ)
- y-coordinate: 0 + x * t * sin(θ)
Using Geometry to Find Time
To find the time when the lion catches the deer, we need to set the lion's coordinates equal to the deer's coordinates:
- For the x-coordinate: 100 - x * t * cos(θ) = 0
- For the y-coordinate: x * t * sin(θ) = yt
From the first equation, we can solve for cos(θ):
cos(θ) = (100 - 0) / (x * t)
Substituting this into the second equation gives us:
sin(θ) = yt / (x * t)
Using the Pythagorean Identity
Since sin²(θ) + cos²(θ) = 1, we can substitute our expressions for sin(θ) and cos(θ):
(yt / (x * t))² + ((100 - 0) / (x * t))² = 1
After simplifying, we can solve for t:
y²t² + 100² = x²t²
(x² - y²)t² = 100²
t² = 100² / (x² - y²)
t = 100 / √(x² - y²)
Final Result
The time t when the lion catches the deer is given by:
t = 100 / √(x² - y²)
This formula shows that the time it takes for the lion to catch the deer depends on the difference in their speeds and the initial distance between them. The faster the lion runs compared to the deer, the quicker it will catch up.
