Question

Two lumps of putty are moving in opposite directions, each one having a speed of \(30.0 \mathrm{m} / \mathrm{s} .\) They collide and stick together. After the collision the combined lumps are at rest. If the mass of each lump was $1.00 \mathrm{kg}$ before the collision, and no energy is lost to the environment, what is the change in mass of the system due to the collision? (tutorial: colliding bullets)

Short answer

Answer: The change in mass of the system due to the collision is 0.

Step-by-step solution

Recall and write the conservation of momentum formula

The conservation of momentum states that the total momentum of a closed system of objects remains constant, provided no external forces are acting on the system. This can be expressed mathematically as: $$(m_1 v_1 + m_2 v_2)_{initial} = (m_1 v_1 + m_2 v_2)_{final}$$

Set up the initial and final momenta

Initially, both lumps of putty are moving. Therefore, their velocities are 30.0 m/s in opposite directions. Let's set the direction of lump 1 as positive. Then, the initial momentum can be written as: $$m_1 v_1 + m_2 v_2 = 1.00 \times 30 + 1.00 \times (-30) = 0$$ After the collision, the lumps stick together and are at rest. Hence, the final momentum is 0.

Apply the conservation of momentum

Now, equate the initial and final momenta using the conservation of momentum principle. $$(m_1 v_1 + m_2 v_2)_{initial} = (m_1 v_1 + m_2 v_2)_{final} = 0$$ We've already found that the initial momentum is 0, so the equation is satisfied.

Recall and write the conservation of energy formula

The conservation of energy states that the total energy of a closed system remains constant, provided no energy is lost to the environment or gained from the environment. This can be expressed mathematically for kinetic energy as: $$\frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 = \text{constant}$$

Set up the initial and final energies

Initially, both lumps of putty have kinetic energies. Therefore, their initial energies can be written as: $$\frac{1}{2} (1.00) (30)^2 + \frac{1}{2} (1.00) (30)^2 = 2 \times \frac{1}{2} \times 1 \times 900 = 1800\ \mathrm{J}$$ However, after the collision they stick together and come to a complete stop. Therefore, their final energy is 0. Since no energy is lost to the environment, the remaining energy should be converted into internal energy or heat.

Calculate the energy conversion

As we established earlier, the initial energy was \(1800\ \mathrm{J}\). After the collision, there is no kinetic energy, so all of this energy should be converted into internal energy or heat. This causes the putty to heat up and possibly change its mass.

Determine the change in mass

Given that no energy is lost to the environment, the change in mass of the system due to the collision should be 0. This is because the energy change due to the collision is completely accounted for in the form of internal energy or heat. Therefore, there is no change in mass. The answer is: The change in mass of the system due to the collision is 0.