A rescue airplane is flying horizontally at a height of 3.0 km with a speed 272 m/s when it releases a package of emergency food supplies.
How far horizontally in meters does the package travel while it falls before it hits the ground?
To calculate the horizontal distance traveled by the package while it falls, we need to determine the time it takes for the package to hit the ground.
First, let's assume that there is no air resistance acting on the package during its fall, which is a simplification for solving this problem.
The initial vertical velocity of the package is 0 m/s since it is released from the airplane, so we can use the kinematic equation:
h = (1/2) * g * t^2
Where h is the height (3.0 km = 3000 m), g is the acceleration due to gravity (9.8 m/s^2), and t is the time.
Rearranging the equation, we get:
t = sqrt(2h/g)
Plugging in the values:
t = sqrt(2 * 3000 / 9.8) ≈ 24.49 seconds (rounded to two decimal places)
Now that we know the time it takes for the package to fall, we can calculate the horizontal distance using the equation:
distance = velocity * time
The velocity is the horizontal speed of the airplane, which is given as 272 m/s.
distance = 272 * 24.49 ≈ 6662.28 meters
Therefore, the package travels approximately 6662.28 meters horizontally before it hits the ground.
To find the horizontal distance that the package travels while it falls, we need to determine the time it takes for the package to hit the ground.
The time it takes for an object to fall from a particular height can be calculated using the equation for vertical motion:
d = (1/2) * g * t^2
Where:
d is the distance traveled in the vertical direction (in this case, the height of the airplane, which is 3.0 km or 3000 m),
g is the acceleration due to gravity (approximately 9.8 m/s^2), and
t is the time taken to fall.
Rearranging the equation to solve for time:
t^2 = (2 * d) / g
t = sqrt((2 * d) / g)
Substituting the values:
t = sqrt((2 * 3000) / 9.8)
t = sqrt(2 * 306.12)
t ≈ sqrt(612.24)
t ≈ 24.74 seconds
Now that we have the time taken to fall, we can calculate the horizontal distance traveled using the formula:
d = v * t
Where:
d is the horizontal distance traveled,
v is the horizontal velocity of the package, and
t is the time taken to fall.
Given the horizontal velocity of the airplane (272 m/s) is the same as the package's horizontal velocity, the formula becomes:
d = 272 * 24.74
d ≈ 6735.28 meters
Therefore, the package travels approximately 6735.28 meters horizontally before it hits the ground.