A plane drops a hamper of medical supplies
from a height of 4190 m during a practice run
over the ocean. The plane’s horizontal veloc-
ity was 111 m/s at the instant the hamper
was dropped.
What is the magnitude of the overall ve-
locity of the hamper at the instant it strikes
the surface of the ocean? The acceleration of
gravity is 9.8 m/s2 .
Answer in units of m/s
To find the magnitude of the overall velocity of the hamper when it strikes the surface of the ocean, we can use the concept of projectile motion. We need to calculate the vertical and horizontal components of the hamper's velocity separately and then combine them to find the overall velocity.
First, let's calculate the time taken for the hamper to fall from a height of 4190 m. We can use the equation:
h = (1/2) * g * t^2
Where:
h = initial height (4190 m)
g = acceleration due to gravity (9.8 m/s^2)
t = time
Let's rearrange the equation to solve for t:
t^2 = (2 * h) / g
t = sqrt((2 * h) / g)
Substituting the given values:
t = sqrt((2 * 4190) / 9.8)
t ≈ 28.628 s
Now, let's calculate the vertical component of velocity at the instant the hamper strikes the surface of the ocean. We can use the equation:
v_vertical = g * t
Substituting the given values:
v_vertical = 9.8 * 28.628
v_vertical ≈ 282.003 m/s
Next, let's calculate the horizontal component of velocity. The horizontal velocity of the hamper remains constant throughout the motion and is given as 111 m/s.
Finally, we can find the overall velocity by using the Pythagorean theorem, which states that the square of the hypotenuse (overall velocity) is equal to the sum of the squares of the two other sides (vertical and horizontal velocity). Thus:
overall velocity^2 = (v_horizontal)^2 + (v_vertical)^2
Substituting the given values:
overall velocity^2 = (111)^2 + (282.003)^2
overall velocity^2 ≈ 31,392 + 79,484.001
overall velocity^2 ≈ 110,876.001
overall velocity ≈ sqrt(110,876.001)
overall velocity ≈ 332.95 m/s
Therefore, the magnitude of the overall velocity of the hamper when it strikes the surface of the ocean is approximately 332.95 m/s.