A plane drops a hamper of medical supplies from a height of 2560 m during a practice run over the ocean. The plane’s horizontal velocity was 116 m/s at the instant the hamper was dropped.What is the magnitude of the overall velocity of the hamper at the instant it strikes the surface of the ocean? The acceleration of gravity is 9.8 m/s^2
Answer in units of m/s
To find the magnitude of the overall velocity of the hamper at the instant it strikes the surface of the ocean, we can use the principles of projectile motion.
The horizontal velocity of the hamper remains constant since there is no horizontal acceleration. Therefore, the magnitude of the horizontal velocity is the same as the plane's horizontal velocity, which is 116 m/s.
The vertical velocity of the hamper changes due to the acceleration of gravity pulling it downward. We can use the equations of motion to find the vertical velocity at the instant it strikes the surface of the ocean.
The vertical displacement can be found using the equation:
Δy = V_iy * t + (1/2) * a_y * t^2,
where Δy is the vertical displacement, V_iy is the initial vertical velocity, a_y is the acceleration in the vertical direction (due to gravity), and t is the time elapsed.
At the instant the hamper is dropped, the initial vertical velocity is zero (V_iy = 0), and the initial vertical displacement is 2560 m. The acceleration due to gravity is -9.8 m/s^2 (negative because it acts downward). Also, the time taken for the hamper to reach the surface of the ocean can be found using the equation:
t = sqrt(2 * Δy / a_y).
Plugging in the values:
t = sqrt(2 * 2560 m / -9.8 m/s^2) ≈ 16.08 s.
Now, using the value of t, we can find the final vertical velocity using the equation:
V_fy = V_iy + a_y * t.
Again, since V_iy is zero, we have:
V_fy = a_y * t,
V_fy = -9.8 m/s^2 * 16.08 s ≈ -157.98 m/s.
The final vertical velocity is negative because it is directed downward.
Now, to find the magnitude of the overall velocity of the hamper at the instant it strikes the surface of the ocean, we can use the Pythagorean theorem:
Magnitude of the overall velocity = sqrt( horizontal velocity^2 + vertical velocity^2)
= sqrt(116 m/s^2)^2 + (-157.98 m/s^2)^2)
= sqrt(13456 + 24947.6004)
= sqrt(38303.6004)
≈ 195.7 m/s.
Therefore, the magnitude of the overall velocity of the hamper at the instant it strikes the surface of the ocean is approximately 195.7 m/s.
To find the magnitude of the overall velocity of the hamper at the instant it strikes the ocean, we can use the horizontal and vertical components of the velocity.
First, let's find the time it takes for the hamper to reach the ocean surface. We can use the equation for vertical displacement under constant acceleration:
Δy = V_{i_y}t + 0.5at^2
Where:
Δy = vertical displacement (2560 m)
V_{i_y} = initial vertical velocity (0 m/s, as the hamper was dropped vertically downward)
a = acceleration due to gravity (-9.8 m/s^2)
t = time
Rearranging the equation to solve for time:
2560 = 0.5(-9.8)t^2
t^2 = -2560 / (-4.9)
t^2 = 523.28
t ≈ 22.88 s
The time it takes for the hamper to reach the ocean surface is approximately 22.88 seconds.
Now, let's find the vertical component of the velocity at the instant it strikes the ocean. We can use the equation for velocity under constant acceleration:
V_f = V_{i_y} + at
Where:
V_f = final vertical velocity (what we want to find)
V_{i_y} = initial vertical velocity (0 m/s)
a = acceleration due to gravity (-9.8 m/s^2)
t = time (22.88 s)
V_f = 0 + (-9.8)(22.88)
V_f ≈ -224.18 m/s (negative sign indicates downward direction)
The vertical component of the velocity at the instant the hamper strikes the ocean is approximately -224.18 m/s.
Lastly, let's find the horizontal component of the velocity. We know that the horizontal velocity remains constant throughout the motion, so it is simply 116 m/s.
Now, to find the magnitude of the overall velocity, we can use the Pythagorean theorem:
V = √(V_x^2 + V_y^2)
where V_x is the horizontal component of the velocity and V_y is the vertical component of the velocity.
V = √(116^2 + (-224.18)^2)
V ≈ √(13456 + 50244.7524)
V ≈ √63700.7524
V ≈ 252.384 m/s
Therefore, the magnitude of the overall velocity of the hamper at the instant it strikes the surface of the ocean is approximately 252.384 m/s.