Question

Резиновый шарик свободно падает и, пролетев 5 метров, ударяется о крышу движущегося вверх со скоростью 0,5 м/с лифта. Считая удар абсолютно упругим, определить, на какую высоту поднимется шарик после отскока. Шахту лифта считать бесконечно высокой. Считать, что ускорение свободного падения g=10м/с2

Answer 1

V²=2gh
v²=2•10•5=100 
v=10 м/с скорость мяча перед ударом относительно земли.
Скорость относительно лифта v₂=10+0,5=10,5 м/с
v₂²=2gh₂
h₂=v₂²/2g=110,25/20≈5,5 м

Answer 2

Problem Analysis

We are given a scenario where a rubber ball is freely falling and travels a distance of 5 meters before colliding with the roof of an elevator moving upward with a velocity of 0.5 m/s. The collision is assumed to be perfectly elastic. We need to determine the height to which the ball will rebound after the collision.

Solution

To solve this problem, we can use the principle of conservation of mechanical energy. The total mechanical energy of the system (ball + elevator) is conserved before and after the collision.

The mechanical energy of an object can be defined as the sum of its kinetic energy and potential energy. In this case, the potential energy is due to the height of the ball above the ground, and the kinetic energy is due to the motion of the ball.

Before the collision, the ball has gravitational potential energy and no kinetic energy. After the collision, the ball will have kinetic energy and potential energy due to its new height.

Let's break down the solution into steps:

1. Calculate the initial potential energy of the ball before the collision. 2. Calculate the initial kinetic energy of the ball before the collision. 3. Calculate the total mechanical energy before the collision. 4. Calculate the final kinetic energy of the ball after the collision. 5. Calculate the final potential energy of the ball after the collision. 6. Calculate the final height of the ball after the collision.

Step 1: Calculate the initial potential energy of the ball before the collision

The initial potential energy of the ball is given by the formula:

Potential Energy = mass * gravitational acceleration * height

Given that the mass of the ball is not provided, we can assume a mass of 1 kg for simplicity. The gravitational acceleration is given as 10 m/s^2.

Initial Potential Energy = 1 kg * 10 m/s^2 * 5 m

Step 2: Calculate the initial kinetic energy of the ball before the collision

The initial kinetic energy of the ball is zero because it is not moving before the collision.

Initial Kinetic Energy = 0

Step 3: Calculate the total mechanical energy before the collision

The total mechanical energy before the collision is the sum of the initial potential energy and the initial kinetic energy.

Total Mechanical Energy before collision = Initial Potential Energy + Initial Kinetic Energy

Step 4: Calculate the final kinetic energy of the ball after the collision

Since the collision is perfectly elastic, the total mechanical energy is conserved. Therefore, the final kinetic energy of the ball after the collision is equal to the initial kinetic energy.

Final Kinetic Energy = Initial Kinetic Energy = 0

Step 5: Calculate the final potential energy of the ball after the collision

The final potential energy of the ball after the collision is given by the formula:

Potential Energy = mass * gravitational acceleration * height

Given that the mass of the ball is not provided, we can assume a mass of 1 kg for simplicity. The gravitational acceleration is given as 10 m/s^2.

Final Potential Energy = 1 kg * 10 m/s^2 * height

Step 6: Calculate the final height of the ball after the collision

The total mechanical energy after the collision is the sum of the final potential energy and the final kinetic energy. Since the total mechanical energy is conserved, we can equate it to the total mechanical energy before the collision.

Total Mechanical Energy after collision = Final Potential Energy + Final Kinetic Energy

Equating the total mechanical energy before and after the collision:

Total Mechanical Energy before collision = Total Mechanical Energy after collision

Initial Potential Energy + Initial Kinetic Energy = Final Potential Energy + Final Kinetic Energy

Substituting the values we calculated earlier:

1 kg * 10 m/s^2 * 5 m + 0 = 1 kg * 10 m/s^2 * height + 0

Simplifying the equation:

50 m = 10 m/s^2 * height

Solving for height:

height = 50 m / 10 m/s^2

height = 5 m

Answer

After the collision, the rubber ball will rebound to a height of 5 meters.

Conclusion

In this problem, we used the principle of conservation of mechanical energy to determine the height to which a rubber ball will rebound after colliding with the roof of a moving elevator. By calculating the initial and final potential energies and kinetic energies, we were able to equate the total mechanical energy before and after the collision and solve for the final height of the ball.