Here is the Hw question:
An archerfish hunts by dislodging an unsuspecting insect from its resting place with a stream of water expelled from the fish’s mouth. Suppose the archerfish squirts water with a speed of 2.3 m/s at an angle of 50 degrees above the horizontal, aiming for a beetle on a leaf that is 3 cm above the water’s surface. At what horizontal distance (in cm) should the archerfish be from the beetle to minimize the time the beetle has to react?
I am not sure where to start any help would be greatly appreciated. Thanks!
To solve this problem, we need to analyze the motion of the water stream and the path of the insect. We can then determine the optimal horizontal distance for the archerfish to minimize the time the beetle has to react.
1. Start by breaking down the initial velocity of the water stream into its horizontal and vertical components.
- The speed of the water stream is 2.3 m/s, and the angle above the horizontal is 50 degrees.
- The horizontal component of the velocity is given by Vx = V * cos(θ).
- The vertical component of the velocity is given by Vy = V * sin(θ).
2. Determine how long it takes for the water stream to reach the beetle's height.
- The initial vertical position of the beetle is 3 cm above the water's surface.
- The vertical displacement (Δy) traveled by the water stream in time (t) is given by Δy = Vy * t + (1/2) * g * t^2, where g is the acceleration due to gravity.
- Set Δy equal to 3 cm (0.03 m) and solve for t.
3. Calculate the horizontal displacement (Δx) traveled by the water stream in time (t).
- The horizontal displacement is given by Δx = Vx * t.
4. The horizontal distance between the archerfish and the beetle is the value of Δx when t is minimized.
- Use calculus (take the derivative and find the critical point) to find the time t at which the horizontal displacement is minimized.
5. Convert the horizontal distance to centimeters as requested in the question.
By following these steps, you should be able to solve the problem and determine the optimal horizontal distance for the archerfish to minimize the time the beetle has to react.