An airplane with a speed of 85.4 m/s is climbing upward at an angle of 37.6 ° with respect to the horizontal. When the plane's altitude is 520 m, the pilot releases a package. (a) Calculate the distance along the ground, measured from a point directly beneath the point of release, to where the package hits the earth. (b) Relative to the ground, determine the angle of the velocity vector of the package just before impact.
To solve this problem, we can break it down into two parts:
(a) Calculating the distance along the ground:
To determine the distance along the ground, we need to find the time it takes for the package to hit the earth. We can do this by using the vertical motion of the package.
To begin, let's analyze the vertical motion of the package. We know that the initial vertical velocity of the package is the same as the vertical component of the airplane's velocity, which is given by:
Vy = V * sin(theta)
= 85.4 m/s * sin(37.6°)
Next, let's find the time it takes for the package to hit the ground. We can use the following equation:
y = voy * t + (1/2) * a * t^2
In this case, y is the initial altitude of the plane (520 m), voy is the initial vertical velocity we found earlier, a is the acceleration due to gravity (-9.8 m/s^2 since the package is moving upwards), and t is the time.
Plugging in the values, we get:
520 m = (85.4 m/s * sin(37.6°)) * t + (1/2) * (-9.8 m/s^2) * t^2
This is a quadratic equation, so we can solve it using the quadratic formula. Once we find the value of t, we can use it to calculate the horizontal distance covered by the package along the ground using:
x = Vx * t
Where Vx is the horizontal component of the airplane's velocity, given by:
Vx = V * cos(theta)
= 85.4 m/s * cos(37.6°)
(b) Determining the angle of the velocity vector:
To find the angle of the velocity vector just before impact, we can use the tangent of the angle. The tangent of an angle is given by the ratio of the vertical component of velocity to the horizontal component of velocity:
tan(theta) = Vy / Vx
theta = atan(Vy / Vx)
Plugging in the values of Vy and Vx, we can calculate the angle theta.
You can now plug in the given values and solve the equations to find the answers.