Question

По горизонтальній поверхні рухається тіло масою 5 кг. На його шляху знаходиться нерухоме тіло масою 2,5 кг. Після центрального абсолютно пружного удару друге тіло набуває кінетичної енергії 5 Дж. Знайти кінетичну енергію першого тіла до і після удару

Answer 1

Problem Analysis

We have a horizontal collision between two bodies: one with a mass of 5 kg and the other with a mass of 2.5 kg. The collision is central and perfectly elastic. We need to find the kinetic energy of the first body before and after the collision.

Solution

To solve this problem, we can use the principle of conservation of momentum and the principle of conservation of kinetic energy.

According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be expressed as:

m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final

where m1 and m2 are the masses of the first and second bodies respectively, v1_initial and v2_initial are their initial velocities, and v1_final and v2_final are their final velocities.

According to the principle of conservation of kinetic energy, the total kinetic energy before the collision is equal to the total kinetic energy after the collision. Mathematically, this can be expressed as:

0.5 * m1 * v1_initial^2 + 0.5 * m2 * v2_initial^2 = 0.5 * m1 * v1_final^2 + 0.5 * m2 * v2_final^2

where v1_initial and v2_initial are the initial velocities of the first and second bodies respectively, and v1_final and v2_final are their final velocities.

We are given that the second body acquires a kinetic energy of 5 J after the collision. Therefore, we can write:

0.5 * m2 * v2_final^2 = 5

We need to find the kinetic energy of the first body before and after the collision. Let's solve the problem step by step.

Step 1: Finding the initial velocities

Since the second body is initially at rest, its initial velocity (v2_initial) is 0. We can substitute this value into the momentum conservation equation:

m1 * v1_initial + m2 * 0 = m1 * v1_final + m2 * v2_final

Simplifying the equation, we get:

m1 * v1_initial = m1 * v1_final

This implies that the initial and final velocities of the first body are equal. Let's denote this common velocity as v_initial.

Step 2: Finding the final velocities

Using the principle of conservation of momentum, we can rewrite the equation as:

m1 * v_initial = m1 * v_final

Since the masses of the first body are the same, we can cancel them out:

v_initial = v_final

This means that the initial and final velocities of the first body are equal.

Step 3: Finding the kinetic energy of the first body before the collision

Using the principle of conservation of kinetic energy, we can substitute the values we found into the equation:

0.5 * m1 * v_initial^2 + 0.5 * m2 * 0^2 = 0.5 * m1 * v_final^2 + 0.5 * m2 * v2_final^2

Simplifying the equation, we get:

0.5 * m1 * v_initial^2 = 0.5 * m1 * v_final^2 + 0.5 * m2 * v2_final^2

Since v_initial = v_final, we can rewrite the equation as:

0.5 * m1 * v_initial^2 = 0.5 * m1 * v_initial^2 + 0.5 * m2 * v2_final^2

Substituting the given values, we have:

0.5 * 5 * v_initial^2 = 0.5 * 5 * v_initial^2 + 0.5 * 2.5 * v2_final^2

Simplifying the equation, we get:

0.5 * 5 * v_initial^2 - 0.5 * 5 * v_initial^2 = 0.5 * 2.5 * v2_final^2

0 = 0.5 * 2.5 * v2_final^2

Since the left side of the equation is 0, the right side must also be 0. Therefore, v2_final = 0.

Step 4: Finding the kinetic energy of the first body after the collision

Using the principle of conservation of kinetic energy, we can substitute the values we found into the equation:

0.5 * m1 * v_initial^2 = 0.5 * m1 * v_final^2 + 0.5 * m2 * v2_final^2

Substituting the given values, we have:

0.5 * 5 * v_initial^2 = 0.5 * 5 * v_final^2 + 0.5 * 2.5 * 0^2

Simplifying the equation, we get:

0.5 * 5 * v_initial^2 = 0.5 * 5 * v_final^2

Since v_initial = v_final, we can rewrite the equation as:

0.5 * 5 * v_initial^2 = 0.5 * 5 * v_initial^2

The left and right sides of the equation are equal, so the kinetic energy of the first body after the collision is the same as its kinetic energy before the collision.

Conclusion

The kinetic energy of the first body before and after the collision is the same.