To solve the problem of determining the angle at which the man should throw the apple to reach the boat, we can use the principles of projectile motion. The key here is to understand how the initial velocity, angle of projection, and horizontal distance interact in this scenario.
Understanding the Problem
The man is sitting on the shore, and he is 5.5 meters away from the center of the boat, which is 1.0 meter away from the shore. This means the total horizontal distance he needs to cover is 5.5 meters. The apple is thrown with an initial speed of 10 m/s. Our goal is to find the angles of projection that allow the apple to land in the boat.
Key Equations of Projectile Motion
In projectile motion, the horizontal and vertical components of the motion can be analyzed separately. The horizontal distance (range) covered by the projectile can be calculated using the formula:
- Range (R) = (v0² * sin(2θ)) / g
Where:
- v0 = initial velocity (10 m/s)
- θ = angle of projection
- g = acceleration due to gravity (approximately 9.81 m/s²)
Calculating the Range
We can rearrange the formula to find the angle of projection that allows the apple to reach the boat:
- R = 5.5 m
- v0 = 10 m/s
- g = 9.81 m/s²
Substituting these values into the range formula gives us:
5.5 = (10² * sin(2θ)) / 9.81
Rearranging this equation allows us to isolate sin(2θ):
sin(2θ) = (5.5 * 9.81) / 100
Calculating this yields:
sin(2θ) ≈ 0.53955
Finding the Angles
Now, we need to find the angles that correspond to this sine value. Using the inverse sine function:
2θ = arcsin(0.53955)
Calculating this gives:
2θ ≈ 32.5 degrees
Thus, θ ≈ 16.25 degrees. However, since the sine function is periodic, we also consider the second solution:
2θ = 180 - 32.5 = 147.5 degrees, leading to θ ≈ 73.75 degrees.
Evaluating the Options
Now, let’s evaluate the provided angle ranges:
- a) 15 degrees to 18.5 degrees - This range does not include our calculated angles.
- b) 15 degrees to 75 degrees - This range includes both 16.25 degrees and 73.75 degrees.
- c) 18.5 degrees to 71.5 degrees - This range does not include our calculated angles.
- d) 71.5 degrees to 75 degrees - This range only includes 73.75 degrees.
Conclusion
The most suitable option that encompasses the angles we calculated is b) 15 degrees to 75 degrees. This range allows for both possible angles of projection that would enable the apple to reach the boat successfully.
