Question

William tell must split the apple atop his son's head from a distance of 24 m. when he aims directly at the apple, the arrow is horizontal. at what angle must he aim it to hit the apple if the arrow is shot at a speed of 25.1 m/s? (find the smaller of two possible angles.)

Answer 1

Let x = the angle of elevation for shooting the arrow.

Assume
g = 9.8 m/s²
No wind resistance
The vertical launch velocity is 25.1 sin(x) m/s
The horizontal velocity is 25.1 cos(x) m/s

The time of flight is
24/[25.1 cos(x)] s = 0.9562 sec(x) s

Therefore
0.5*[0.9562 sec(x)]*(9.8)  = 25.1 sin(x)
4.6854 = 25.1* sin(x)cos(x)
sin(2x) = 0.3733
2x = sin⁻¹ 0.3733 = 21.92 deg
x = 10.96 deg

Answer: 11 degrees (nearest integer)

Answer 2

Answer : Angle is $\theta=11^0$

Explanation :

It is given that,

Horizontal range, R = 24 m

Velocity, v = 25.1 m/s

We have to find the angle with which he hit the apple.

We know that horizontal range is given by the relation as :

$R=\dfrac{v^2\ sin2\theta}{g}$

Putting all values in above equation we get:

$sin2\theta=\dfrac{R\ g}{v^2}$

$sin2\theta=\dfrac{24\ m\times 9.8\ m/s^2}{(25.1\ m)^2}$

$sin2\theta=0.373$

$\theta=10.95^0$

or

$\theta=11^0$    

So, the angle must he aim it to hit the apple is $\theta=11^0$.