An arrow is shot from a height of 1.4 m toward a cliff of height H. It is shot with a velocity of 26 m/s at an angle of 60° above the horizontal. It lands on the top edge of the cliff 3.4 s later.
(a) What is the height of the cliff (in m)?
(b) What is the maximum height (in m) reached by the arrow along its trajectory?
(c) What is the arrow's impact speed (in m/s) just before hitting the cliff?
27.1
To solve this problem, we can use the equations of projectile motion. Let's break it down step by step:
Step 1: Find the vertical component of the initial velocity (Vy) using the given angle and magnitude of the velocity.
Given: Velocity (v) = 26 m/s, Angle (θ) = 60°
Using the equation Vy = v * sin(θ), we can calculate:
Vy = 26 m/s * sin(60°) = 22.516 m/s
Step 2: Find the time it takes for the arrow to reach the top edge of the cliff (t1).
Given: Initial height (y0) = 1.4 m, Vertical displacement (Δy) = H - y0, where H is the height of the cliff.
Using the formula Δy = Vy * t1 - (0.5 * g * t1^2), where g is the acceleration due to gravity (approximately 9.8 m/s^2) and t1 is the time taken to reach the top edge of the cliff.
Substituting the values, we get:
H - 1.4 = 22.516 * t1 - 0.5 * 9.8 * t1^2
Step 3: Find the total time of flight (T) of the arrow.
Given: Total time of flight (T) = 3.4 s
Since the arrow lands on the top edge of the cliff after 3.4 seconds, the time it takes for the arrow to reach the top edge of the cliff (t1) is half of the total time of flight.
Hence, t1 = T / 2 = 3.4 s / 2 = 1.7 s.
Step 4: Solve for H in the equation obtained in Step 2.
H - 1.4 = 22.516 * t1 - 0.5 * 9.8 * t1^2
Simplifying the equation and solving for H, we get:
H = 1.4 + 22.516 * t1 - 0.5 * 9.8 * t1^2
Step 5: Calculate the maximum height reached by the arrow (hmax) using the equation hmax = y0 + Vy^2 / (2 * g).
Given: Initial height (y0) = 1.4 m, Vertical component of the initial velocity (Vy) = 22.516 m/s, Acceleration due to gravity (g) = 9.8 m/s^2.
Substituting the values, we get:
hmax = 1.4 + (22.516)^2 / (2 * 9.8)
Step 6: Calculate the horizontal component of the initial velocity (Vx) using the equation Vx = v * cos(θ).
Given: Velocity (v) = 26 m/s, Angle (θ) = 60°
Using the equation, Vx = v * cos(θ), we can calculate:
Vx = 26 m/s * cos(60°) = 13 m/s
Step 7: Find the horizontal displacement (x) of the arrow.
Using the formula x = Vx * T, where T is the total time of flight, we can calculate:
x = 13 m/s * 3.4 s = 44.2 m
Step 8: Calculate the final vertical component of the velocity just before hitting the cliff (Vy_final).
Given: Vy_final = Vy - g * T.
Substituting the values, we get:
Vy_final = 22.516 m/s - 9.8 m/s^2 * 3.4 s
Now we have all the step-by-step solutions:
(a) Height of the cliff (H) = 1.4 + 22.516 * 1.7 - 0.5 * 9.8 * (1.7)^2
(b) Maximum height reached by the arrow (hmax) = 1.4 + (22.516)^2 / (2 * 9.8)
(c) Arrow's impact speed just before hitting the cliff (Vy_final) = 22.516 m/s - 9.8 m/s^2 * 3.4 s
To find the answers to these questions, we can use the equations of projectile motion.
Let's break down the problem step by step.
Step 1: Determine the time taken to reach the maximum height.
The time taken to reach the maximum height can be found using the equation:
t = (V * sinθ) / g
where:
- t is the time taken
- V is the initial velocity (26 m/s)
- θ is the launch angle (60°)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
Plugging in the given values, we can calculate the time taken to reach the maximum height.
t = (26 * sin(60°)) / 9.8
t ≈ 1.88 s
Step 2: Calculate the maximum height reached by the arrow.
The maximum height reached can be found using the equation:
h_max = (V^2 * sin^2θ) / (2g)
Plugging in the given values, we can calculate the maximum height.
h_max = (26^2 * sin^2(60°)) / (2 * 9.8)
h_max ≈ 22.3 m
So, the maximum height reached by the arrow is approximately 22.3 m.
Step 3: Calculate the time taken to reach the cliff top.
The total time taken can be calculated by adding the time to reach the maximum height (found in Step 1) to the time taken for the arrow to fall from the maximum height.
total_time = t + t
total_time = 1.88 s + 1.88 s
total_time = 3.76 s
Since the total time taken to reach the top of the cliff is given as 3.4 s, we can set up an equation:
3.4 = 3.76 - t_cliff
where t_cliff is the time taken for the arrow to fall from the maximum height to the top of the cliff.
Solving for t_cliff:
t_cliff = 3.76 - 3.4
t_cliff = 0.36 s
Step 4: Calculate the height of the cliff.
The height of the cliff can be found using the equation:
H = V * cosθ * t_cliff
Plugging in the given values, we can calculate the height of the cliff.
H = 26 * cos(60°) * 0.36
H ≈ 12.1 m
So, the height of the cliff is approximately 12.1 m.
Step 5: Calculate the impact speed of the arrow.
The impact speed of the arrow just before hitting the cliff can be found using the equation:
V_impact = V * cosθ
Plugging in the given values, we can calculate the impact speed.
V_impact = 26 * cos(60°)
V_impact ≈ 13 m/s
So, the impact speed of the arrow just before hitting the cliff is approximately 13 m/s.
To summarize:
(a) The height of the cliff is approximately 12.1 m.
(b) The maximum height reached by the arrow is approximately 22.3 m.
(c) The impact speed of the arrow just before hitting the cliff is approximately 13 m/s.