an object falls from a plane flying horizontally at an altitude of 40,000 ft at 500 mi/h. How long will it take to hit the ground?
To determine the time it takes for an object to hit the ground, we need to analyze the motion in the vertical direction. We can use the equation of motion for free fall:
h = 0.5 * g * t^2
where h is the vertical distance traveled (the altitude of the plane in this case), g is the acceleration due to gravity (approximately 32.2 ft/s^2), and t is the time.
Given that the altitude is 40,000 ft, we can set h = 40,000 ft and solve for t.
40,000 = 0.5 * 32.2 * t^2
Dividing both sides by 0.5 * 32.2, we get:
t^2 = 40,000 / (0.5 * 32.2)
t^2 = 40,000 / 16.1
t^2 = 2,484.472
Taking the square root of both sides, we find:
t = sqrt(2,484.472)
t ≈ 49.84 s
Therefore, it will take approximately 49.84 seconds for the object to hit the ground.
To find the time it takes for the object to hit the ground, we can use the equations of motion and consider two dimensions: horizontal and vertical.
First, let's focus on the vertical motion of the object. We will use the equation of motion:
h = ut + (1/2)gt^2
Where:
h = height (40,000 ft)
u = initial vertical velocity (0 ft/s, as the object is dropped)
g = acceleration due to gravity (32.2 ft/s^2)
t = time
Substituting the known values into the equation:
40,000 = 0 * t + (1/2) * 32.2 * t^2
Now, let's solve for 't':
To do this, we can rearrange the equation to get a quadratic equation in the form of 'at^2 + bt + c = 0'.
16.1t^2 = 40,000
Dividing both sides by 16.1:
t^2 = 40,000 / 16.1
t^2 ≈ 2484
Taking the square root of both sides:
t ≈ sqrt(2484)
t ≈ 49.84
Since time cannot be negative, we ignore the negative root. The time it takes for the object to reach the ground is approximately 49.84 seconds.
Therefore, it will take approximately 49.84 seconds for the object to hit the ground.
At 40,000 ft. air resistance becomes important, as the object will likely reach terminal velocity and greatly increases the time it takes to reach the ground. Also, the horizontal velocity comes into play as the terminal velocity caused by air resistance is actually a terminal speed, so it's the resultant that counts.
The shape of the object is not specified, so any calculation of the terminal velocity remains an estimate with assumptions.
If you have not done air-resistance and terminal velocities, read on.
Neglecting air resistance, the horizontal velocity of the plane does not count, and you can calculate the time required as if it were a free fall over short distances.
Let
vi=initial velocity (0)
t=time in seconds,
g=acceleration due to gravity, 32.2 f/s^2
S=vertical distance travelled = 40,000 ft
S=vi*t+(1/2)gt^2
40000=0*t + (1/2)*32.2*t^2
Solve for t to get:
t^2=2*40000/32.2=2484.47
t=50 sec. approx.
(remember: this is by ignoring air-resitance)