7. A stuntwoman drives a car off a ramp at the top of a parking desk. The end of the ramp is at the edge of the $22 \mathrm{~m}$ high parking deck, and the ramp is angled upward $30^{\circ}$. The stuntwoman is going $30 \mathrm{~m} / \mathrm{s}$ when she hits the ramp. (a) (5 points) Draw a motion diagram of the car, beginning when it drives off the parking garage, and ending when it hits the ground. (b) (5 points) Draw a pictorial representation of the problem, with drawings of the car at the beginning and end of the motion, and any points in between when the character of the motion changes. Establish a coordinate system. Define symbols for the position, velocity, acceleration, and time at each of your drawings. (c) $\left(2^{1} / 2\right.$ points) Make a table of known quantities, and a table of unknown quantities. (d) $\left(2 \frac{1}{2}\right.$ points) What are the $x$ and $y$ components of the car's initial velocity? (e) (10 points) How long does it take the car to reach the ground? What is the horizontal distance from the base of the parking deck to the point where the car hits the ground?
Solution
Step 1 ：First, we need to find the x and y components of the car's initial velocity. We can use the following equations:
Step 2 ：\(v_x = v \cos(\theta)\)
Step 3 ：\(v_y = v \sin(\theta)\)
Step 4 ：Using the given values, we have:
Step 5 ：\(v_x = 30 \cos(30^\circ) \approx 25.98 \, \text{m/s}\)
Step 6 ：\(v_y = 30 \sin(30^\circ) \approx 15.00 \, \text{m/s}\)
Step 7 ：\boxed{\text{(d)} \, v_x \approx 25.98 \, \text{m/s}, \, v_y \approx 15.00 \, \text{m/s}}
Step 8 ：Now, we can use the kinematic equations to find the time it takes for the car to reach the ground and the horizontal distance it travels.
Step 9 ：\(y = v_{y0}t - \frac{1}{2}gt^2\)
Step 10 ：Solving for time, we get:
Step 11 ：\(t \approx 4.14 \, \text{s}\)
Step 12 ：Next, we can find the horizontal distance using the equation:
Step 13 ：\(x = v_{x0}t\)
Step 14 ：\(x \approx 25.98 \times 4.14 \approx 107.59 \, \text{m}\)
Step 15 ：\boxed{\text{(e)} \, t \approx 4.14 \, \text{s}, \, x \approx 107.59 \, \text{m}}
