An archer standing a horizontal distance d=50m away from a tree
sees an apple hanging from the tree at h=12m above the ground. The
archer chooses an arrow and prepares to shoot. The arrow is initially 1.5m
above the ground. Just as the archer shoots the arrow with a speed of
75m/s, the apple breaks off and falls straight down. A person of height
2.0m is standing directly underneath the apple. The arrow pierced the
apple. Ignore air resistance.
(a) What angle did the archer aim the arrow at?
(b) How high above the person's head did the arrow hit the apple?
enter your answer in m
To solve this problem, we can use projectile motion equations. Let's break it down step by step:
Step 1: Determine the time it takes for the apple to fall from the tree.
Using the equation for free fall:
h = 0.5 * g * t^2
where h is the initial height (12m) and g is the acceleration due to gravity (9.8m/s^2).
Rearranging the equation to solve for time (t):
t = sqrt(2h / g)
Substituting the values:
t = sqrt(2 * 12 / 9.8) = sqrt(24 / 9.8) ≈ 1.38 seconds
Step 2: Calculate the horizontal distance the apple travels during this time.
The horizontal distance the apple travels is given by:
d = v * t
where d is the distance, v is the horizontal velocity, and t is the time calculated in Step 1.
Substituting the values:
d = 75 * 1.38 ≈ 103.5 meters
Step 3: Find the angle at which the archer aimed the arrow.
The horizontal distance traveled by the arrow is the same as that of the apple (103.5 meters). By considering the velocity of the arrow and the time taken, we can find the angle using trigonometry.
Let θ be the angle of projection.
Using the equation for horizontal velocity:
v_x = v * cos(θ)
Rearranging the equation to solve for cos(θ):
cos(θ) = v_x / v
cos(θ) = d / (v * t)
Substituting the values:
cos(θ) = 103.5 / (75 * 1.38)
cos(θ) ≈ 1.32
Note that cos(θ) should be less than or equal to 1, but here it is greater. This means there is no angle that allows the arrow to reach the apple. Hence, in this scenario, it is not possible for the arrow to pierce the apple.
We do not need to calculate the height above the person's head where the arrow would hit the apple since it is not feasible in this case.