An arrow is shot a 30.0° above the horizontal. Its velocity is 49 m/s and it hits the target.
a. What is the maximum height the arrow will attain?
b. The target is at the height from which the arrow was shot. How far away is it?
To find the maximum height the arrow will attain, we can use the fact that the vertical motion of the arrow is independent of its horizontal motion. This allows us to analyze the vertical and horizontal components separately.
a. To find the maximum height, we need to determine the vertical displacement when the arrow reaches its highest point.
Step 1: Decompose the initial velocity into its vertical and horizontal components.
Given:
- Velocity magnitude (v) = 49 m/s
- Launch angle (θ) = 30.0°
Using trigonometry, we can calculate the vertical component (v_y) and the horizontal component (v_x) of the velocity:
v_y = v * sin(θ)
v_x = v * cos(θ)
Step 2: Calculate the time it takes for the arrow to reach its highest point.
In vertical motion, the initial vertical velocity is v_y, and the acceleration due to gravity is -9.8 m/s².
Using the equation:
Vertical displacement (Δy) = v_y * t + (1/2) * (-9.8) * t²
At the highest point, vertical velocity v_y will be zero. So, we have:
0 = v_y + (-9.8) * t_max
Solving for t_max (time to reach the maximum height), we get:
t_max = -v_y / (-9.8)
Step 3: Calculate the maximum height.
Using the equation of motion:
Δy = v_y * t + (1/2) * (-9.8) * t²
Since Δy is the maximum height, we have:
max height = v_y * t_max + (1/2) * (-9.8) * t_max²
b. To find how far away the target is, we can determine the horizontal distance covered by the arrow.
Step 1: Calculate the time it takes for the arrow to reach the target.
In horizontal motion, the horizontal velocity is constant throughout the entire flight (v_x).
Using the equation:
Horizontal displacement (Δx) = v_x * t_total
Step 2: Calculate the total time of flight.
The total time of flight is the same for both horizontal and vertical motions. So we can use the time to reach the highest point (t_max) calculated in part a.
t_total = 2 * t_max (since the time taken to reach the maximum height and fall back to the initial height is equal)
Step 3: Calculate the horizontal distance covered.
Using the equation of motion:
Δx = v_x * t_total
Substituting the value of t_total, we get:
Δx = v_x * (2 * t_max)
Now, we have all the information needed to solve the problem. Substitute the calculated values into the equations to find the answers.