Question

С крыши 50 этажного дома выкинули ботинок, его начальная скрость по координате х составляла 10м/с На каком этаже он окажется через 5 секунд. Высота всего здания 200 метров. Найдите координату х и у

Answer 1

Problem Analysis

We have a shoe thrown from the roof of a 50-story building with an initial velocity of 10 m/s in the x-coordinate direction. We need to determine the floor on which the shoe will land after 5 seconds. The total height of the building is 200 meters. We also need to find the x and y coordinates of the shoe.

Solution

To solve this problem, we need to consider the motion of the shoe in both the x and y directions separately.

# Motion in the x-direction

Since the shoe is thrown horizontally, there is no acceleration in the x-direction. Therefore, the velocity in the x-direction remains constant at 10 m/s throughout the motion. We can use the formula `distance = velocity * time` to find the horizontal distance traveled by the shoe.

The horizontal distance traveled by the shoe can be calculated as follows: ``` distance_x = velocity_x * time ``` where `distance_x` is the horizontal distance traveled by the shoe, `velocity_x` is the initial velocity in the x-direction (10 m/s), and `time` is the time taken (5 seconds).

# Motion in the y-direction

In the y-direction, the shoe is subject to the force of gravity, which causes it to accelerate downwards. We can use the equations of motion to find the vertical distance traveled by the shoe.

The equation for the vertical distance traveled by an object under constant acceleration is given by: ``` distance_y = initial_velocity_y * time + (1/2) * acceleration_y * time^2 ``` where `distance_y` is the vertical distance traveled by the shoe, `initial_velocity_y` is the initial velocity in the y-direction (0 m/s as the shoe is thrown horizontally), `acceleration_y` is the acceleration due to gravity (-9.8 m/s^2), and `time` is the time taken (5 seconds).

# Finding the floor and coordinates

To find the floor on which the shoe will land, we need to determine the number of floors the shoe has passed. Since each floor is 4 meters high (200 meters / 50 floors), we can divide the vertical distance traveled by the shoe by the height of each floor to find the floor number.

The floor number can be calculated as follows: ``` floor_number = distance_y / floor_height ``` where `floor_number` is the floor on which the shoe will land, `distance_y` is the vertical distance traveled by the shoe, and `floor_height` is the height of each floor (4 meters).

To find the x-coordinate of the shoe, we can use the horizontal distance traveled by the shoe. Since the shoe is thrown horizontally, the x-coordinate remains constant throughout the motion.

To find the y-coordinate of the shoe, we can use the vertical distance traveled by the shoe. The y-coordinate will be the height of the building minus the vertical distance traveled by the shoe.

Let's calculate the values.

Calculation

Given: - Initial velocity in the x-direction (velocity_x) = 10 m/s - Time taken (time) = 5 seconds - Total height of the building (building_height) = 200 meters - Height of each floor (floor_height) = building_height / number_of_floors = 200 meters / 50 floors = 4 meters - Acceleration due to gravity (acceleration_y) = -9.8 m/s^2

Using the equations mentioned above, we can calculate the values as follows:

1. Calculate the horizontal distance traveled by the shoe: ``` distance_x = velocity_x * time = 10 m/s * 5 s = 50 meters ```

2. Calculate the vertical distance traveled by the shoe: ``` distance_y = initial_velocity_y * time + (1/2) * acceleration_y * time^2 = 0 m/s * 5 s + (1/2) * (-9.8 m/s^2) * (5 s)^2 = -122.5 meters ```

3. Calculate the floor on which the shoe will land: ``` floor_number = distance_y / floor_height = -122.5 meters / 4 meters = -30.625 floors ``` Since the floor number cannot be negative, we can round it down to the nearest whole number to get the floor on which the shoe will land. Therefore, the shoe will land on the 31st floor.

4. Calculate the x-coordinate of the shoe: The x-coordinate remains constant throughout the motion and is equal to the horizontal distance traveled by the shoe, which is 50 meters.

5. Calculate the y-coordinate of the shoe: ``` y_coordinate = building_height - distance_y = 200 meters - (-122.5 meters) = 322.5 meters ```

Answer

The shoe will land on the 31st floor of the building. The x-coordinate of the shoe is 50 meters and the y-coordinate is 322.5 meters.

Please note that the negative value for the vertical distance traveled indicates that the shoe has fallen below the starting point on the roof of the building.