91Ó°ÊÓ

In physics, momentum is a key concept that describes the quantity of motion an object possesses. It depends on two variables: mass and velocity. Momentum is given by the formula \( p = mv \), where \( m \) is the mass and \( v \) is the velocity. For two colliding objects, the principle of momentum conservation states that the total momentum before the collision is equal to the total momentum after the collision.

In the original exercise, we calculated the initial momentum of both objects using their respective masses and velocities. The object with a mass of \( 25.0 \text{ g} \) and velocity \( 20.0 \text{ cm/s} \) has an initial momentum of \( p_{1i} = (25.0 \text{ g})(20.0 \text{ cm/s}) \). The second object, with mass \( 10.0 \text{ g} \) and velocity \( 15.0 \text{ cm/s} \), has an initial momentum \( p_{2i} = (10.0 \text{ g})(15.0 \text{ cm/s}) \).

• Total initial momentum \( p_i = p_{1i} + p_{2i} \)
• Conservation equation: \( p_i = p_f \)

After the collision, because it is an elastic one, these momentum values help us form an equation that assists in finding the velocities of both objects post-collision.

Kinetic energy refers to the energy an object has due to its motion, and is calculated using the formula \( KE = \frac{1}{2}mv^2 \). In elastic collisions, just like momentum, the total kinetic energy of the system is conserved. This means the total kinetic energy before the collision must equal that after the collision.

In the problem, calculating the initial kinetic energies is straightforward. For the first object, we have \( KE_{1i} = \frac{1}{2} \cdot 25.0 \text{ g} \cdot (20.0 \text{ cm/s})^2 \). The second object has an initial kinetic energy \( KE_{2i} = \frac{1}{2} \cdot 10.0 \text{ g} \cdot (15.0 \text{ cm/s})^2 \).

• Total initial kinetic energy \( KE_i = KE_{1i} + KE_{2i} \)
• Conservation equation: \( KE_i = KE_f \)

Ensuring kinetic energy conservation helps us verify that the collision remains elastic, and it aids in constructing a second equation needed to calculate the final velocities of the objects post-impact.

Velocity is the speed of an object in a particular direction, and it plays a crucial role in collisions. In this exercise, initial velocities were given and are essential in calculating both momentum and kinetic energy.

After the collision, our job is to find the final velocities of the two objects. We have two unknowns I—both final velocities—but also two essential equations from momentum and kinetic energy conservation. This gives us enough information to solve the problem.

• Initial velocities: \( v_{1i} = 20.0 \text{ cm/s} \), \( v_{2i} = 15.0 \text{ cm/s} \)
• Final velocities: \( v_{1f} \text{ and } v_{2f} \)
• Two equations: \( p_i = p_f \) and \( KE_i = KE_f \)

Using these conservation laws, the final velocities \( v_{1f} \) and \( v_{2f} \) are determined, solving the equations based on the provided values.