To determine how far from the dog the suitcase will land, we need to analyze the situation using some principles of physics, specifically projectile motion. The problem can be broken down into two main components: the horizontal motion and the vertical motion of the suitcase after it is dropped from the airplane.
Step 1: Initial Velocity Components
First, let's calculate the horizontal and vertical components of the airplane's velocity. The airplane is flying at a velocity of 90 m/s at an angle of 23 degrees above the horizontal. We can use trigonometric functions to find these components:
- Horizontal Velocity (Vx): Vx = V * cos(θ) = 90 m/s * cos(23°)
- Vertical Velocity (Vy): Vy = V * sin(θ) = 90 m/s * sin(23°)
Calculating these components:
- Vx ≈ 90 * 0.9205 ≈ 82.85 m/s
- Vy ≈ 90 * 0.3907 ≈ 35.16 m/s
Step 2: Time of Fall
Next, we need to find out how long it takes for the suitcase to fall 114 meters to the ground. We can use the following kinematic equation for vertical motion:
y = Vy * t + 0.5 * g * t²
Here, y is the vertical distance (114 m), Vy is the initial vertical velocity (35.16 m/s), g is the acceleration due to gravity (approximately 9.81 m/s²), and t is the time in seconds. Plugging in the values, we need to rearrange the equation:
114 = 35.16 * t + 0.5 * 9.81 * t²
This is a quadratic equation in the form of:
0 = 4.905t² + 35.16t - 114
Step 3: Solving the Quadratic Equation
We can solve this quadratic equation using the quadratic formula:
t = (-b ± √(b² - 4ac)) / 2a
In our case, a = 4.905, b = 35.16, c = -114:
Calculating the discriminant:
- b² - 4ac = (35.16)² - 4 * 4.905 * (-114)
- ≈ 1238.1856 + 2236.68 ≈ 3474.8656
Now, substituting into the quadratic formula:
t = [−35.16 ± √(3474.8656)] / (2 * 4.905)
Calculating the square root and the two possible t values, we find:
t ≈ 1.78 seconds (only the positive root is physically meaningful).
Step 4: Horizontal Distance Calculation
Now that we have the time of fall, we can calculate how far the suitcase travels horizontally during that time:
Horizontal Distance = Vx * t
Substituting the values:
Distance = 82.85 m/s * 1.78 s ≈ 147.67 meters.
Final Result
The suitcase will land approximately 147.67 meters away from the dog. This distance is measured horizontally from the position directly below where the suitcase was dropped.
