An airplane with a speed of 85.3 m/s is climbing upward at an angle of 45.0 ° with respect to the horizontal. When the plane's altitude is 587 m, the pilot releases a package. (a) Calculate the distance along the ground, measured from a point directly beneath the point of release, to where the package hits the earth. (b) Relative to the ground, determine the angle of the velocity vector of the package just before impact.
a. Vo = 85.3m/s[45o]
Xo = 85.3*Cos45 = 60.32 m/s.
Yo = 85.3*sin45 = 60.32 m/s.
h = 0.5g*t^2 = 587 m.
4.9t^2 = 587
t^2 = 119.8
Tf = 10.95 s. = Fall time.
Dx = Xo * Tf = 60.32m/s * 10.95s=660 m.
b. Y = Yo + g*Tf = 0 + 9.8*10.95 = 107.3 m/s.
Tan A = Y/Xo = 107.3/60.32 = 1.77822
A = 60.65o
To solve this problem, we can break it down into two parts: calculating the horizontal distance traveled by the package and finding the angle of the velocity vector just before impact.
(a) Calculating the horizontal distance traveled by the package:
To find the horizontal distance, we need to determine how long it takes for the package to hit the ground and then multiply that time by the horizontal component of the plane's velocity.
Step 1: Calculate the time of flight.
Since the package is released from the plane in mid-air, we can use the formula for time of flight (t) from kinematics:
h = v₀yt + (1/2)gt²
where h is the initial height, v₀y is the initial vertical component of the velocity, g is the acceleration due to gravity (approximately 9.8 m/s²), and t is the time of flight.
Given:
h = 587 m
v₀y = v₀sinθ (vertical component of the initial velocity)
v₀ = 85.3 m/s (speed of the airplane)
θ = 45.0° (angle of ascent)
Calculating v₀y:
v₀y = v₀ * sin(θ)
= 85.3 * sin(45.0°)
Now, let's calculate t using the above equation:
587 = (85.3 * sin(45.0°))t + (1/2)(9.8)t²
We can rearrange this equation to solve for t using the quadratic formula:
0.5(9.8)t² + (85.3 * sin(45.0°))t - 587 = 0
Solving this quadratic equation will give us the value of t.
Step 2: Calculate the horizontal distance.
Now that we have the time of flight, we can calculate the horizontal distance traveled by the package using the formula:
d = v₀x * t
where v₀x is the horizontal component of the initial velocity.
Calculating v₀x:
v₀x = v₀ * cos(θ)
= 85.3 * cos(45.0°)
Now, we can calculate d by multiplying v₀x by t.
(b) Determining the angle of the velocity vector just before impact:
To find the angle of the velocity vector just before impact, we can use the concept of the relative motion of the package with respect to the ground.
Since the horizontal component of the velocity remains constant throughout the flight, the angle of the velocity vector just before impact will be the same as the angle of the plane's ascent, which is 45.0°.