To determine the minimum and maximum angles at which the man can throw the apple to successfully land it in the boat, we can use the principles of projectile motion. The key here is to analyze the horizontal and vertical components of the throw, given the distance to the boat and the speed of the throw.
Understanding the Problem
The man is sitting 5.5 meters away from the center of a 1-meter long boat. This means that the edges of the boat are located at 5 meters (5.5m - 0.5m) and 6 meters (5.5m + 0.5m) from the man. Therefore, the apple must land within this range of 5 to 6 meters. The speed at which he can throw the apple is 10 m/s.
Breaking Down the Motion
In projectile motion, the horizontal distance (range) covered by the projectile can be calculated using the formula:
- Range (R) = (v^2 * sin(2θ)) / g
Where:
- v = initial velocity (10 m/s)
- g = acceleration due to gravity (approximately 9.81 m/s²)
- θ = angle of projection
Calculating the Range
We need to find the angles that allow the apple to land between 5 meters and 6 meters. First, we can rearrange the range formula to solve for the angle:
- R = (v^2 * sin(2θ)) / g
- sin(2θ) = (R * g) / v^2
Finding the Minimum Angle
For the minimum distance (5 meters):
- sin(2θ) = (5 * 9.81) / (10^2)
- sin(2θ) = 0.4905
Now, we can find the angle:
- 2θ = arcsin(0.4905)
- 2θ ≈ 29.3°
- θ ≈ 14.65°
Finding the Maximum Angle
For the maximum distance (6 meters):
- sin(2θ) = (6 * 9.81) / (10^2)
- sin(2θ) = 0.5886
Now, we can find this angle as well:
- 2θ = arcsin(0.5886)
- 2θ ≈ 35.9°
- θ ≈ 17.95°
Final Results
In summary, the man can successfully throw the apple into the boat at angles between approximately:
- Minimum angle: 14.65°
- Maximum angle: 17.95°
These angles ensure that the apple lands within the 1-meter length of the boat, given the distance of 5.5 meters from the man to the center of the boat. This analysis illustrates the fascinating interplay of physics and projectile motion in real-world scenarios.
