Question

Камень брошенный вертикально вверх достиг высоты 15 м и упал на то же место. Найти модуль перемещения и пройденный путь?

Answer 1

Problem Analysis

We are given that a stone is thrown vertically upwards and reaches a height of 15 m before falling back to the same spot. We need to find the displacement and the distance traveled by the stone.

Solution

To find the displacement, we need to calculate the difference between the initial and final positions of the stone. Since the stone falls back to the same spot, the displacement will be zero.

To find the distance traveled by the stone, we need to calculate the total path length covered by the stone. This can be calculated by adding the distance traveled upwards and downwards.

Let's calculate the distance traveled upwards and downwards separately.

Distance Traveled Upwards

When the stone is thrown vertically upwards, it will follow a parabolic path. The distance traveled upwards can be calculated using the formula:

distance_upwards = initial_velocity * time + (1/2) * acceleration * time^2

In this case, the stone is thrown vertically upwards, so the initial velocity is positive and the acceleration due to gravity is negative. The time taken to reach the maximum height can be calculated using the formula:

time_to_reach_max_height = initial_velocity / acceleration

The initial velocity can be calculated using the formula:

initial_velocity = sqrt(2 * acceleration * height)

where height is the maximum height reached by the stone.

Substituting the values, we can calculate the distance traveled upwards.

Distance Traveled Downwards

When the stone falls back to the same spot, it will follow a similar path as when it was thrown upwards. The distance traveled downwards can be calculated using the same formula as for the distance traveled upwards.

Substituting the values, we can calculate the distance traveled downwards.

Total Distance Traveled

The total distance traveled by the stone is the sum of the distance traveled upwards and the distance traveled downwards.

Now, let's calculate the displacement and the distance traveled using the given information.

Calculation

Given: - Maximum height reached by the stone: 15 m

Using the formula for initial velocity, we can calculate the initial velocity:

initial_velocity = sqrt(2 * acceleration * height)

Substituting the values:

initial_velocity = sqrt(2 * (-9.8 m/s^2) * 15 m) = 10.98 m/s

Using the formula for time to reach the maximum height, we can calculate the time taken:

time_to_reach_max_height = initial_velocity / acceleration

Substituting the values:

time_to_reach_max_height = 10.98 m/s / (-9.8 m/s^2) = -1.12 s

Note: The negative sign indicates that the stone is moving in the opposite direction of the positive direction.

Using the formula for distance traveled upwards, we can calculate the distance traveled upwards:

distance_upwards = initial_velocity * time + (1/2) * acceleration * time^2

Substituting the values:

distance_upwards = 10.98 m/s * (-1.12 s) + (1/2) * (-9.8 m/s^2) * (-1.12 s)^2 = 6.16 m

Using the same formula, we can calculate the distance traveled downwards:

distance_downwards = initial_velocity * time + (1/2) * acceleration * time^2

Substituting the values:

distance_downwards = 10.98 m/s * (-1.12 s) + (1/2) * (-9.8 m/s^2) * (-1.12 s)^2 = -6.16 m

The displacement is the difference between the initial and final positions, which is zero in this case.

The total distance traveled is the sum of the distance traveled upwards and the distance traveled downwards:

total_distance = distance_upwards + distance_downwards = 6.16 m + (-6.16 m) = 0 m

Answer

The module of displacement is 0 m and the total distance traveled is 0 m.

Please let me know if I can help you with anything else.