An airplane with a speed of 74.5 m/s is climbing upward at an angle of 64.8 degrees with respect to the horizontal. When the plane's altitude is 687 m, the pilot releases the 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.

Vertical speed = 74.5 sin 64.8 = 67.40961541

Extra height reached = 4544.05625 / (2 x 9.8) = 231.8396046

Total height reached = 231.8396046 + 687 = 918.8396046 m

Time to reach max height = 74.5/9.8 = 7.602040816 s

Time to fall 919 m = square root (2 x 918.8396046 / 9.8) = 13.69373166 s

Total time in air = 21.29577248 seconds

Horizontal speed = 74.5 cos 64.8 = 31.72055722 m/s

Horizontal distance = 31.72055722 x 21.29577248 = 675.5137695 m

Vertical speeds at ground = 9.8 x 13.69373166 = 134.1985703 m/s

Angle to the ground velocity = tan^-1 (134.1985703 / 31.72055722) = 76.70106771 degrees

For part A my answer is 134.1985703 m/s
For part B my answer is 76.70106771 degrees
However, the answers are incorrect. What am I doing wrong? Can you please help. Thanks.

In order to calculate the correct answers, let's go through the problem step by step.

First, let's calculate the vertical component of the airplane's velocity using the given speed and angle:

Vertical velocity = speed * sin(angle) = 74.5 m/s * sin(64.8°)

Vertical velocity = 63.35 m/s (rounded to two decimal places)

Next, we need to calculate the time it takes for the package to reach its maximum height. This can be done by dividing the vertical velocity by the acceleration due to gravity:

Time to reach maximum height = vertical velocity / acceleration due to gravity = 63.35 m/s / 9.8 m/s^2

Time to reach maximum height ≈ 6.47 seconds (rounded to two decimal places)

Now, let's calculate the maximum height reached by the airplane:

Maximum height = vertical velocity * time to reach maximum height = 63.35 m/s * 6.47 s

Maximum height ≈ 410.63 meters (rounded to two decimal places)

We can now find the total time of flight by adding the time to reach maximum height to the time it takes for the package to fall from the maximum height to the ground:

Total time of flight ≈ 6.47 s + 6.47 s ≈ 12.94 seconds (rounded to two decimal places)

To find the horizontal distance traveled by the package, we need to multiply the horizontal component of the airplane's velocity by the total time of flight:

Horizontal distance = speed * cos(angle) * total time of flight

Horizontal distance = 74.5 m/s * cos(64.8°) * 12.94 s

Horizontal distance ≈ 411.33 meters (rounded to two decimal places)

For part A, the correct answer is approximately 411.33 meters.

Now let's calculate the angle of the velocity vector of the package just before impact. We can use the vertical velocity and horizontal velocity components at the ground to find the angle:

Angle = arctan(vertical velocity / horizontal velocity)

Angle = arctan(63.35 m/s / (74.5 m/s * cos(64.8°)))

Angle ≈ 47.35 degrees (rounded to two decimal places)

For part B, the correct answer is approximately 47.35 degrees.