An Alaskan rescue plane traveling 49 m/s
drops a package of emergency rations from
a height of 191 m to a stranded party of
explorers.
The acceleration of gravity is 9.8 m/s2.
Where does the package strike the ground
relative to the point directly below where it
was released?
Answer in units of m.
What is the horizontal component of the velocity just before it hits?
Answer in units of m/s.
What is the vertical component of the velocity
just before it hits? (Choose upward as the
positive vertical direction)
Answer in units of m/s.
To solve this problem, we can use the equations of motion to find the position and velocity of the package just before it hits the ground.
1. Determine the time it takes for the package to fall from a height of 191 m. We can use the equation:
h = (1/2) * g * t^2
where h is the height, g is the acceleration due to gravity, and t is the time. Rearranging the equation to solve for t, we have:
t = sqrt(2h/g)
Substituting the given values, we have:
t = sqrt(2 * 191 m / 9.8 m/s^2) = 8.78 seconds
2. Find the horizontal distance traveled by the package during this time. The horizontal distance can be calculated using the equation:
d = v * t
where d is the distance, v is the horizontal velocity, and t is the time. Since there is no horizontal acceleration, the horizontal velocity remains constant throughout the motion. Therefore, the horizontal component of the velocity just before it hits is the same as the initial horizontal velocity of the plane, which is 49 m/s. Substituting the values, we have:
d = 49 m/s * 8.78 s = 430.22 meters
So, the package strikes the ground 430.22 meters away from the point directly below where it was released.
3. Determine the vertical component of the velocity just before it hits. The vertical component of the velocity can be calculated using the equation:
v = g * t
where v is the velocity, g is the acceleration due to gravity, and t is the time. Substituting the values, we have:
v = 9.8 m/s^2 * 8.78 s = 86.04 m/s
Since upward is defined as the positive vertical direction, the vertical component of the velocity just before it hits is -86.04 m/s, indicating a downward direction.
Therefore, the answers to the given questions are:
1. The package strikes the ground 430.22 meters away from the point directly below where it was released.
2. The horizontal component of the velocity just before it hits is 49 m/s.
3. The vertical component of the velocity just before it hits is -86.04 m/s.