An archer shoots an arrow at a 72.0 m distant target, the bull's-eye of which is at same height as the release height of the arrow.
(a) At what angle must the arrow be released to hit the bull's-eye if its initial speed is 36.0 m/s? (Although neglected here, the atmosphere provides significant lift to real arrows.)
Please help me !
you know the range is given by
R = v^2/g sin2θ
so, plugging in your numbers,
36^2/9.8 sin2θ = 72
To find the angle at which the arrow must be released to hit the bull's-eye, we can use the projectile motion equations.
Let's denote the angle of release as θ, the initial speed as v0, and the horizontal distance to the target as d. In this case, d is given as 72.0 m.
The projectile motion equations in the horizontal (x) and vertical (y) directions are as follows:
Horizontal equation: x = v0 * cos(θ) * t
Vertical equation: y = v0 * sin(θ) * t - (1/2) * g * t^2
In the horizontal direction, the time (t) taken by the arrow to reach the target is the same as the time taken to reach the maximum height, which is obtained when the vertical velocity becomes zero during the flight. Thus, we can set the vertical equation equal to zero to find the time at maximum height:
0 = v0 * sin(θ) * t - (1/2) * g * t^2
Next, we can solve this quadratic equation for t:
(1/2) * g * t^2 = v0 * sin(θ) * t
(1/2) * g * t = v0 * sin(θ)
t = (2 * v0 * sin(θ)) / g
Substituting this value of t into the horizontal equation, we get:
x = v0 * cos(θ) * [(2 * v0 * sin(θ)) / g]
Simplifying further, we have:
x = (2 * v0^2 * cos(θ) * sin(θ)) / g
Now, let's substitute the known values into this equation:
x = (2 * (36.0 m/s)^2 * cos(θ) * sin(θ)) / 9.8 m/s^2
x = (2 * 1296 m^2/s^2 * cos(θ) * sin(θ)) / 9.8 m/s^2
x = (2592 m^2/s^2 * cos(θ) * sin(θ)) / 9.8 m/s^2
To hit the bull's-eye, the horizontal distance (x) should be equal to the given 72.0 m:
(2592 m^2/s^2 * cos(θ) * sin(θ)) / 9.8 m/s^2 = 72.0 m
Simplifying this equation will require trigonometric identities and algebraic manipulation. However, solving it directly may not have a simple algebraic solution. In such cases, numerical methods or approximation techniques, like using a graphing calculator or computer software, can be employed to find the value of θ.