Robin Hood has to shoot an arrow at an apple that sits on a wall 400 meters away, at a height of 40 meters. He must shoot at an angle of 60 degrees. What must the initial velocity of the arrow be to hit the target? Assume he's shooting laying down on the ground, making the arrow's inital height to be 0, and use -9.8 m/s² as the value of gravity.

Question

Robin Hood has to shoot an arrow at an apple that sits on a wall 400 meters away, at a height of 40 meters. He must shoot at an angle of 60 degrees. What must the initial velocity of the arrow be to hit the target? Assume he's shooting laying down on the ground, making the arrow's inital height to be 0, and use -9.8 m/s² as the value of gravity.

Answer 1

I'll assume he's lying on the ground... The arrow's height is

y = tanθ x - 4.9/(v cosθ)^2 x^2
so, we need

400√3 - 19.6*400^2/v^2 = 40
v = 69.309 m/s

y = √3 x - 4.9/(69.309/2)^2 x^2
see the graph at
http://www.wolframalpha.com/input/?i=plot+y%3D%E2%88%9A3+x+-+4.9%2F(69.309%2F2)%5E2+x%5E2,+y%3D40

Answer 2

Thanks for the help. But, he is LAYING down on the ground!!!!!!

Answer 3

To determine the initial velocity of the arrow required to hit the target, we can use the following equations of motion:

1. Horizontal motion equation:
Displacement (horizontal) = initial velocity (horizontal) * time

2. Vertical motion equation:
Displacement (vertical) = (initial velocity (vertical) * time) + (0.5 * gravity * time^2)

We can break down the initial velocity into its horizontal and vertical components.

Given:
- Displacement (horizontal) = 400 meters
- Displacement (vertical) = 40 meters
- Launch angle = 60 degrees
- Initial vertical velocity = 0 m/s (since Robin Hood is shooting from ground level)
- Gravity = -9.8 m/s² (negative since it acts downwards)

Step 1: Breaking down the initial velocity.
Initial velocity (horizontal) = initial velocity * cos(angle)
Initial velocity (vertical) = initial velocity * sin(angle)

Step 2: Solving for time.
In horizontal motion, the horizontal displacement is equal to initial velocity (horizontal) multiplied by time.
400 = initial velocity * cos(60) * time
time = 400 / (initial velocity * cos(60))

Step 3: Solving for initial velocity.
In vertical motion, the vertical displacement is equal to initial velocity (vertical) multiplied by time, plus half the gravity multiplied by time squared.
40 = (initial velocity * sin(60) * time) + (0.5 * -9.8 * time^2)

Substituting the value of time from Step 2:
40 = (initial velocity * sin(60) * (400 / (initial velocity * cos(60)))) + (0.5 * -9.8 * (400 / (initial velocity * cos(60)))^2)

Now, we can solve this equation numerically to find the suitable initial velocity for Robin Hood to hit the target.