An airplane with a speed of 94.9 m/s is climbing upward at an angle of 63.2 ° with respect to the horizontal. When the plane's altitude is 743 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 answer this question, we can break it down into smaller parts and use basic kinematic equations. Let's solve it step by step:
Step 1: Calculate the time it takes for the package to hit the ground.
In this case, we can use the vertical motion equation, which relates the displacement, initial velocity, time, and acceleration:
(1) Δy = v0y * t + (1/2) * a * t^2
We know the initial vertical velocity (v0y = 94.9 m/s * sin(63.2°)), and the acceleration in this case is due to gravity (a = -9.8 m/s^2). The displacement (Δy) is the altitude (743 m), and we need to solve for time (t).
Plug in the known values into equation (1):
743 m = (94.9 m/s * sin(63.2°)) * t + (1/2) * (-9.8 m/s^2) * t^2
Step 2: Solve the quadratic equation to find t.
Rearrange the equation to put it in the form of a quadratic equation:
(1/2) * (-9.8 m/s^2) * t^2 + (94.9 m/s * sin(63.2°)) * t - 743 m = 0
Now, solve this quadratic equation using the quadratic formula or any other method. You will find two possible solutions for t. However, you should choose the positive solution since time cannot be negative.
Step 3: Calculate the horizontal distance traveled by the package.
To calculate the horizontal distance traveled (Δx), we can use the horizontal motion equation:
(2) Δx = v0x * t
where v0x is the initial horizontal velocity of the package. This can be calculated using the initial velocity (94.9 m/s) and the angle of ascent (63.2°):
v0x = 94.9 m/s * cos(63.2°)
Now, plug in the known values of v0x and t into equation (2) to find Δx, which represents the horizontal distance.
Step 4: Calculate the angle of the velocity vector just before impact.
To find the angle of the velocity vector just before impact, we can use trigonometry. Since we already know the horizontal and vertical components of the velocity vector, we can use the tangent function:
(3) tan(θ) = v0y / v0x
Solve equation (3) for θ to find the angle.
Now you have the step-by-step approach to solve this problem, so go ahead and plug in the values to get the final answers for both parts (a) and (b).