if you were to aim right at the bulls-eye, the arrow would fall under gravity and drop below the intended target. If you intend to hit a bulls-eye it is necessary that you aim slightly above it. Suppose the arrow speed is the same (70 m/s) and the target is 10 meters away. What launch angle, Θ, is needed so that the arrow hits the bulls-eye?
please just tell the answer
A = 0.573 Degrees.
To determine the launch angle (Θ) needed for the arrow to hit the bulls-eye, we can use the equations of projectile motion and consider the effect of gravity.
The key idea is that the vertical component of the arrow's velocity will cause it to fall due to gravity while the horizontal component will keep it moving forward. We need to find the launch angle that provides the right combination of forward velocity and vertical drop to hit the target.
Let's break down the problem step by step:
Step 1: Identify known values:
- Arrow speed (v): 70 m/s
- Distance to the target (d): 10 meters
- Acceleration due to gravity (g): 9.8 m/s² (assuming no air resistance)
Step 2: Decompose the velocity into horizontal and vertical components:
The horizontal component of velocity (Vx) remains constant throughout the entire flight because there is no horizontal acceleration. So, Vx = v * cos(Θ).
The vertical component of velocity (Vy) changes due to the acceleration of gravity. Initially, Vy = v * sin(Θ), but it changes throughout the flight.
Step 3: Determine the time of flight (t):
Using kinematic equations, we can find the time it takes for the arrow to travel the horizontal distance to the target:
d = Vx * t
t = d / Vx
Step 4: Determine the vertical displacement (h):
Using a kinematic equation for vertical motion, we can find the vertical displacement of the arrow by substituting the known values:
h = Vy * t + (1/2) * (-g) * t²
h = (v * sin(Θ)) * t - (1/2) * g * t²
Step 5: Set up the equation to hit the bulls-eye:
The arrow hits the bulls-eye when the vertical displacement (h) is zero because it means the arrow has fallen back to the same height as the target.
Substituting the expressions for t, Vx, and h from previous steps, we get:
0 = (v * sin(Θ)) * (d / Vx) - (1/2) * g * (d / Vx)²
Simplifying the equation will give us the relationship between the launch angle (Θ) and the other known values. Solving for Θ may require the use of numerical methods or an iterative process.
Bear in mind that the calculation assumes ideal conditions with no air resistance and that the target is at the same height as the launch point. In reality, these factors may affect the required launch angle.
Overall, the launch angle (Θ) can be determined by using the equations of projectile motion and considering the effect of gravity on the arrow's trajectory.