A helicopter is ascending vertically with a speed of 3.00 m/s. At a height of 70 m above the Earth, a package is dropped from a window. How much time (in seconds) does it take for the package to reach the ground? [Hint: The package's initial speed equals the helicopter's.]
This question was asked a few days ago, but perhaps not by you. The equation for height (above the ground, in meters) vs time (in seconds) is
y = 70 + 3 t - (g/2) t^2
Use g = 9.8 m/s^2 and solve for the time when y = 0. The quadratic equation will give you two answers. Take the positive one.
To find the time it takes for the package to reach the ground, we can use the equation of motion for vertical motion:
h = (1/2) * g * t^2
where h is the vertical distance, g is the acceleration due to gravity, and t is the time.
In this case, the package is dropped from a height of 70 m, so we can set h to be -70 m (negative because it's moving downwards). The acceleration due to gravity, g, is approximately 9.8 m/s^2.
Substituting these values into the equation, we get:
-70 = (1/2) * 9.8 * t^2
Rearranging the equation, we have:
t^2 = (-70 * 2) / 9.8
t^2 = -140 / 9.8
t^2 ≈ -14.29
Since time cannot be negative, this result is invalid. It means that the package will never reach the ground if it's simply dropped from the helicopter.
However, if we assume that the package has some initial downward velocity (in addition to the helicopter's upward velocity), then we can calculate the time it takes for the package to reach the ground.
Let's assume the package has an initial downward velocity of 3.00 m/s (equal to the helicopter's ascending velocity). Now we can use the equation of motion:
h = (1/2) * g * t^2 + v_i * t
where v_i is the initial velocity.
Substituting the values:
-70 = (1/2) * 9.8 * t^2 + 3.00 * t
Rearranging the equation, we have:
0 = (1/2) * 9.8 * t^2 + 3.00 * t + 70
This equation is a quadratic equation. We can solve it by applying the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
From the quadratic equation, we have:
a = (1/2) * 9.8
b = 3.00
c = 70
Substituting these values into the quadratic formula, we can find the values of t.