Question

5. William Tell is said to have shot an apple off his son’s head with an arrow. If the arrow was shot with an initial speed of 58.0 m/s and the boy was 27.0 m away, at what launch angle above the horizontal did Bill aim the arrow? (Assume that the arrow and apple are initially at the same height above the ground. Hint: sin2θ = 2sinθcosθ)

Answer 1

Answer: 2.254 °

Explanation:

To solve this, first we have to take in account the height of the apple and the arrow's initial position, they are the same. Because of this, then for the arrow to hit the apple, the arrow's reached his highest point at the same time is was on half the distance from William to his son, 13.5m.

Also, conservation of energy tell us that at the highest point the Y-Compound of the Velocity is equals to Zero, because the effects of gravity. This is, because velocity as a Vector can be divided on his dimensional compounds. Which are:

Vy = V * sinФ

Vx= V * cosФ

We can try and calculate this time on which the arrow was on half travel with the following equations:

For the X-Axis, it a movement which it continuous and uniform, so the following equations applies:

Velocity (V) = Distance (D) / Time (T)

T = D / V

We reemplace values for Distance and the equation of Velocity X-Compound

T = 13.5m / (V * cosФ)

T = 13.5m / (58m/s  *  cosФ)

For the Y- Axis, we can think a movement under the effect of earth's gravity. So this equation can work in our case:

Final Velocity (Vf) = Initial Velocity (Vo) + (Acceleration (a) * Time (T))

Where, the Final Velocity at the middle point is Zero, the Initial Velocity is V Y-Compound and Acceleration is the gravity

0 = Vy +  (-G * T)

Gravity is a negative value, because it is against or movement.

We try to know the Time on this equation

T = Vy / G

T = (V * sinФ) / G

T= (58m/s  *  cosФ) / 9.81$m/s^{2}$

Because it is the same Time in which the Y-Compound becomes zero and the Arrow reached half of the trip we can say then:

T = 13.5m / (58m/s  *  cosФ) = (58m/s  *  sinФ) / 9.81$m/s^{2}$

Now we try to search for the angle:

13.5m * 9.81$m/s^{2}$  =  (58m/s  *  cosФ)  * (58m/s  *  sinФ)

132.435 = 3364 * cosФ * sinФ

cosФ * sinФ = 0.0393

With our hint, we know that:  sin2θ = 2sinθcosθ

However, because there is no two next to our cosФ * sinФ we need to do the following:

$\frac{2}{2}  * cosФ  * sinФ$ = 0.0393

2* cosФ  * sinФ = 0.0393 * 2

sin (2Ф) = 0.0786

The angle can be know, by Angle's Property:

Arcsin (Sin 2Ф) =  2Ф

Soo we run the following operation in our equation:

Arcsin (0.0786) = 4.508° = 2Ф

Dividing this by two:

Ф = 4.508 / 2

Ф = 2.254 °

2.254 ° being the angle on which William launched his arrow.