Problem 71 - College Physics

Question

In a noninjury chain-reaction accident on a foggy freeway, car 1 (mass of \(2000 \mathrm{~kg}\) ) moving at \(15.0 \mathrm{~m} / \mathrm{s}\) to the right elastically collides with car 2 , initially at rest. The mass of car 2 is \(1500 \mathrm{~kg}\). In turn, car 2 then goes on to lock bumpers (that is, it is a completely inelastic collision) with car \(3,\) which has a mass of \(2500 \mathrm{~kg}\) and was also at rest. Determine the speed of all cars immediately after this unfortunate accident.

Step-by-Step Solution

Answer

Car 1: 6 m/s, Car 2 and Car 3 (together): 4.5 m/s.

Step 1: Elastic Collision between Car 1 and Car 2

In an elastic collision, both momentum and kinetic energy are conserved. First, write the conservation of momentum equation:\[ m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \]Given that car 2 is initially at rest, \( v_{2i} = 0 \), we have:\[ 2000 \times 15 + 1500 \times 0 = 2000 \times v_{1f} + 1500 \times v_{2f} \]This simplifies to:\[ 30000 = 2000 v_{1f} + 1500 v_{2f} \]We'll call this Equation (1).

Step 2: Kinetic Energy Conservation for Car 1 and Car 2

For elastic collisions, kinetic energy is conserved. Write the kinetic energy conservation equation:\[ \frac{1}{2} m_1 v_{1i}^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2 \]Substitute the given values:\[ \frac{1}{2} \times 2000 \times 15^2 = \frac{1}{2} \times 2000 \times v_{1f}^2 + \frac{1}{2} \times 1500 \times v_{2f}^2 \]Simplifying, this becomes:\[ 225000 = 1000 v_{1f}^2 + 750 v_{2f}^2 \]We'll call this Equation (2).

Step 3: Solve Equations for Velocities of Car 1 and Car 2

We have two simultaneous equations from Steps 1 and 2:Equation (1):\[ 30000 = 2000 v_{1f} + 1500 v_{2f} \]Equation (2):\[ 225000 = 1000 v_{1f}^2 + 750 v_{2f}^2 \]By solving these equations, we can determine the velocities:Substitute \(v_{1f} = 15 - \frac{3}{4}v_{2f}\) from Equation (1) into Equation (2). After solving, \(v_{1f} = 6 \text{ m/s}\) and \(v_{2f} = 12 \text{ m/s}\).

Step 4: Inelastic Collision between Car 2 and Car 3

In an inelastic collision, momentum is conserved but not kinetic energy. When car 2 with speed \(v_{2f} = 12 \text{ m/s}\) collides inelastically with car 3 (initially at rest), they move together after collision. Write the momentum conservation equation:\[ m_2 v_{2f} + m_3 v_{3i} = (m_2 + m_3) v_f \]Given \(v_{3i} = 0\):\[ 1500 \times 12 = (1500 + 2500) v_f \]This simplifies to:\[ 18000 = 4000 v_f \]Solving for \(v_f\), we get \(v_f = 4.5 \text{ m/s}\).

Key Concepts

Momentum Conservation

Momentum conservation is a fundamental principle in physics. It's vital in understanding how motion works, especially during collisions. This principle states that within a closed system, the total momentum before a collision is equal to the total momentum after the collision. This holds true regardless of the type of collision, be it elastic or inelastic.

In our exercise, car 1 (2000 kg) collides with car 2 (1500 kg). Initially, car 2 is at rest, so its momentum is zero. By using the momentum conservation equation, we can say that:

• For the elastic collision, the momentum before the collision is simply the momentum of car 1.
• After the collision, both cars have some momentum, and their total must equal the initial momentum of car 1.
• For the inelastic collision where car 2 locks with car 3 (2500 kg, also initially at rest), the combined mass moves with a new velocity. Here, we apply conservation again to find this new velocity.

These applications demonstrate how momentum conservation is crucial in solving collision problems efficiently.

Kinetic Energy Conservation

While momentum is always conserved, kinetic energy conservation occurs only in elastic collisions. This means that the total kinetic energy, which is the energy of motion, remains constant during these collisions.

In the example from the exercise:

• During the elastic collision between car 1 and car 2, we apply the kinetic energy conservation equation to determine their velocities after the collision.
• By equating the total initial kinetic energy with the total final kinetic energy, we derive another equation that helps us solve for the unknown velocities.

It's important to note that in inelastic collisions, like the one between car 2 and car 3, kinetic energy is not conserved. Some of it turns into other forms of energy, like heat or sound.

Understanding when and how to apply kinetic energy conservation is pivotal in accurately predicting post-collision results.

Physics Problem Solving

Solving physics problems can appear daunting at first. However, breaking them into manageable steps can simplify the process considerably. Successful problem solving in physics often involves:

• Identifying known and unknown quantities from a given problem.
• Applying relevant physical principles, such as conservation laws.
• Setting up equations that reflect these laws accurately, as seen in the stepwise resolution of car collision problems.

In our example:

• We first tackled the elastic collision using both momentum and kinetic energy conservation principles. This gave us two equations that we solved simultaneously.
• Next, we approached the inelastic collision focusing on momentum conservation to find the final velocity of the combined mass of cars 2 and 3.

With practice, these strategies become intuitive, enhancing your ability to solve complex physics problems effectively. Always remember to stay organized and check each step for accuracy to master problem-solving in physics.