The speed of an arrow fired from a compound

bow is about 57 m/s. An archer sits astride
his horse and launches an arrow into the air,
elevating the bow at an angle of 54
above the horizontal and 2.3 m above the ground.
What is the arrow's range?

time in air:

hf=ho+vi*sin54*t-4.9t^2

ho=2.3, hf=0
solve that above equation for time t.

Then, distancehorizontal
d=vi*cos54*time

To find the arrow's range, we need to determine the horizontal distance it travels before hitting the ground. We can use the projectile motion equations to solve this problem.

Let's break down the given information:

Initial velocity (v0) = 57 m/s (speed of the arrow)
Launch angle (θ) = 54 degrees (elevation angle of the bow)
Height from the ground (h) = 2.3 m

We can now use the range equation to find the horizontal distance traveled by the arrow:

Range (R) = (v0^2 * sin(2θ)) / g,

where g is the acceleration due to gravity, approximately 9.8 m/s^2.

First, we need to find the value of sin(2θ):

sin(2θ) = sin(2 * 54)
= sin(108)
≈ 0.934

Next, substitute the given values into the range equation:

R = (57^2 * 0.934) / 9.8,
R ≈ 313.26 m.

Therefore, the arrow's range is approximately 313.26 meters.

To determine the arrow's range, we can break down the motion into horizontal and vertical components.

1. Horizontal Component:
Since the arrow is launched at an angle of 54° above the horizontal, we can find the horizontal component of its velocity using the equation:

Vx = V * cos(θ)

where V is the speed of the arrow (57 m/s) and θ is the launch angle (54°).

Vx = 57 m/s * cos(54°)
Vx = 57 m/s * 0.5878
Vx = 33.47 m/s

2. Vertical Component:
The vertical component of the velocity will help us calculate the time the arrow is in the air. We can use the equation:

Vy = V * sin(θ)

Vy = 57 m/s * sin(54°)
Vy = 57 m/s * 0.8090
Vy = 46.05 m/s

3. Time of Flight:
To find the time the arrow is in the air, we can use the vertical component of velocity and the equation:

Vy = gt

where g is the acceleration due to gravity. Assuming g = 9.8 m/s².

46.05 m/s = 9.8 m/s² * t
t = 46.05 m/s / 9.8 m/s²
t = 4.7 s

4. Range:
The range of the arrow can now be calculated using the horizontal component of velocity and the time of flight:

Range = Vx * t
Range = 33.47 m/s * 4.7 s
Range = 157.21 m

Therefore, the arrow's range is approximately 157.21 meters.