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 can break it down into several steps:
Step 1: Find the time it takes for the water stream to reach the beetle's height.
Step 2: Calculate the horizontal distance traveled by the water stream in that time.
Step 3: Determine the horizontal distance between the archerfish and the beetle when the water stream reaches the beetle's height.
Let's start with Step 1:
1. Find the time it takes for the water stream to reach the beetle's height:
- We can use the formula for vertical motion under constant acceleration.
- The initial vertical velocity (v₀) is 2.3 m/s multiplied by the sine of the launch angle (50 degrees).
- The vertical displacement (Δy) is 3 cm, which is equivalent to 0.03 m.
- The acceleration due to gravity (a) is -9.8 m/s² (negative because it acts downward).
- Use the formula: Δy = v₀t + (1/2)at² and solve for t.
Now, let's move to Step 2:
2. Calculate the horizontal distance traveled by the water stream in that time:
- Since we know the time (t) it takes for the water stream to reach the beetle's height, we can use this time to find the horizontal distance traveled.
- The horizontal velocity (v_x) remains constant throughout the motion and is given by v₀ multiplied by the cosine of the launch angle (50 degrees).
- Use the formula: Horizontal Distance = v_x * t.
Finally, we reach Step 3:
3. Determine the horizontal distance between the archerfish and the beetle when the water stream reaches the beetle's height.
- This is the answer we are looking for, as it will minimize the time the beetle has to react.
By following these steps, we can find the horizontal distance at which the archerfish should be from the beetle to minimize the time the beetle has to react.