A package in an airplane moving horizontally at 150m/s is dropped when the altitude is 490m. (a) How long does it take the package to fall to the ground ?
(b)How far forward from the spot over which it was dropped does the package land ? (c) What kind of path does the package follow ?
To find the answers to these questions, we can use the equations of motion under constant acceleration.
(a) To find the time it takes for the package to fall to the ground, we need to find the time it takes for the package to reach a height of 0m.
We can use the equation:
s = ut + (1/2)at^2
Where:
s = displacement (change in height)
u = initial velocity (0 m/s, as the package is dropped)
a = acceleration (due to gravity, approximately 9.8 m/s^2)
t = time taken
Rearranging the equation to solve for t, we have:
t = sqrt((2s) / a)
Substituting the values:
s = 490m
a = 9.8 m/s^2
t = sqrt((2 x 490) / 9.8)
t = sqrt(980 / 9.8)
t = sqrt(100)
t = 10 seconds
So, it takes 10 seconds for the package to fall to the ground.
(b) To find how far forward from the spot over which it was dropped does the package land, we need to find the horizontal distance traveled by the package in 10 seconds.
Since the horizontal velocity of the plane does not affect the vertical motion of the package, the horizontal distance traveled will be given by:
distance = velocity x time
distance = 150 m/s x 10 s
distance = 1500 m
Therefore, the package lands 1500 meters forward from the spot over which it was dropped.
(c) The path followed by the package is a parabolic curve due to the vertical acceleration caused by gravity. This is known as projectile motion.
To answer these questions, we need to use the equations of motion. Let's go step by step:
(a) To find the time it takes for the package to fall, we can use the equation for vertical motion:
s = ut + (1/2)gt^2
Where:
s = vertical distance
u = initial vertical velocity (0 m/s since the package is dropped)
g = acceleration due to gravity (-9.8 m/s^2, assuming no air resistance)
t = time taken
Given:
s = 490m (vertical distance)
Substituting the values into the equation, we have:
490 = 0*t + (1/2)(-9.8)t^2
Simplifying the equation:
490 = -4.9t^2
Rearranging:
t^2 = 490/(-4.9)
t^2 = -100
Since time cannot be negative, there seems to be an issue here. It implies that the package does not reach the ground. Please check the given values or assumptions.
(b) Since the package does not reach the ground in part (a), we cannot determine how far forward it lands. It appears to be a hypothetical scenario.
(c) As for the type of path the package follows, if it falls freely without any horizontal forces acting on it (neglecting air resistance), it follows a straight line vertically downward. If there are any horizontal forces present, it would follow a curved path due to the combined effect of vertical and horizontal motion.
In summary, due to the inconsistency in the given values, we cannot determine the time it takes for the package to fall, how far forward it lands, or the exact path it follows.