William Tell must split the apple atop his son’s head from a distance of 45.1m. When William aims directly at the apple, the arrow is horizontal. At what angle must he aim it to hit the apple if the arrow travels at a speed of 45.956m/s
time of flight:
t=distance/speed=45.1m/(s*cosTheta) where s is the launch speed.
so you have not t in terms of theta.
time in air:
hf=hi+vi*sinTheta*t-4.9t^2 hf=hi=0
so put t in the equationi, you should be able to solve for theta. You may have to use a double angle formula if you dont remember, look it up.
To solve this problem, we can use the principles of projectile motion. The arrow's horizontal and vertical motions are independent of each other. Let's break down the given information:
- The initial speed of the arrow (v) is 45.956 m/s.
- The distance to the apple (d) is 45.1 m.
- The acceleration due to gravity (g) is approximately 9.8 m/s².
To find the angle at which William must aim the arrow, we need to determine its vertical and horizontal components of velocity separately. We'll use the following equations of motion:
Vertical motion equation:
d = v₀t + (1/2)gt²
Horizontal motion equation:
d = v₀t
First, let's calculate the time it takes for the arrow to reach the apple horizontally. We can use the horizontal motion equation:
45.1 m = (45.956 m/s) * t
Solving for t, we find:
t = 45.1 m / 45.956 m/s
t ≈ 0.9815 s
Now, let's calculate the height the arrow reaches vertically. We can use the vertical motion equation and consider that the final displacement in the vertical direction is zero:
0 m = (0 m/s) * t + (1/2)(-9.8 m/s²)t²
Simplifying the equation, we get:
-4.9 m/s² * t² = 0
Since the square of a quantity is only zero if the quantity itself is zero, we find that t = 0 s or t = √(0/4.9). Since t cannot be zero, we have:
t = √(0/4.9)
t = 0 s
This means that during the entire time the arrow is in horizontal motion, it remains at the same vertical level without any change in height.
Now, let's calculate the angle at which William must aim the arrow. We can use the definition of the tangent of an angle:
tan(θ) = (change in vertical position) / (change in horizontal position)
Since there is no change in vertical position, the angle (θ) must be zero degrees. Therefore, William must aim the arrow horizontally to hit the apple.
In summary, William must aim the arrow horizontally (at 0 degrees) to hit the apple when it is positioned at a distance of 45.1 m and the arrow is traveling at a speed of 45.956 m/s.