Problem 42 - College Physics

Question

A \(1200-\mathrm{kg}\) car traveling initially with a speed of \(25.0 \mathrm{~m} / \mathrm{s}\) in an easterly direction crashes into the rear end of a \(9000-\mathrm{kg}\) truck mowing in the same direction at \(20.0 \mathrm{~m} / \mathrm{s}\) (Fig. P6.42). The velocity of the car right after the collision is \(18.0 \mathrm{~m} / \mathrm{s}\) to the east. (a) What is the velocity of the truck right after the collision? (b) How much mechanical energy is lost in the collision? Account for this loss in energy-

Step-by-Step Solution

Answer

(a) The velocity of the truck just after the collision is approximately \(20.4 \,m/s\) to the east. (b) The amount of mechanical energy lost in the collision calculated in the last step is the answer to this part.

Step 1: Calculate Initial Momentum

First, let's calculate the initial momentum of both the car and the truck before the collision. The momentum (p) of an object is given by the product of its mass (m) and velocity (v), i.e., \(p = m \cdot v\). For the car: \(p_{\text{car initial}} = 1200\,kg \cdot 25.0\,m/s = 30000\,kg \cdot m/s\), and for the truck: \(p_{\text{truck initial}} = 9000\, kg \cdot 20.0\, m/s = 180000\, kg \cdot m/s\).

Step 2: Determine Final Momentum

Now we must find the final momentum of the car and the truck after the collision using the same formula. We know that the final velocity of the car is \(18.0\,m/s\), therefore the final momentum of the car is \(p_{\text{car final}} = 1200\,kg \cdot 18.0\,m/s = 21600 \,kg \cdot m/s\). We don't know the final velocity of the truck yet, so we'll call this \(v_{\text{truck final}}\) for now.

Step 3: Apply Conservation of Momentum

The law of conservation of momentum states that the total momentum before a collision is equal to the total momentum after the collision. Therefore, we can set up the following equation: \(p_{\text{car initial}} + p_{\text{truck initial}} = p_{\text{car final}} + (9000\, kg \cdot v_{\text{truck final}})\). Substituting the known values into this equation, we have: \(30000\,kg \cdot m/s + 180000\, kg \cdot m/s = 21600\, kg \cdot m/s + (9000\, kg \cdot v_{\text{truck final}})\). Solving for \(v_{\text{truck final}}\) yields \(v_{\text{truck final}} = 20.4\, m/s\)

Step 4: Calculate Initial and Final Mechanical Energy

Now we need to find the mechanical energy before and after the collision. The mechanical energy is the sum of kinetic energy, which is given by the formula \(KE = 0.5 \cdot m \cdot v^2\). Before the collision, the total kinetic energy is the sum of car’s and truck's kinetic energy: \(KE_{\text{initial}} = 0.5 \cdot 1200\, kg \cdot (25.0\, m/s)^2 + 0.5 \cdot 9000\, kg \cdot (20.0\, m/s)^2 \). Similarly, after the collision, total kinetic energy is summed of car’s and truck's kinetic energy: \(KE_{\text{final}} = 0.5 \cdot 1200\, kg \cdot (18.0\, m/s)^2 + 0.5 \cdot 9000\, kg \cdot (20.4\, m/s)^2 \)

Step 5: Calculate Energy Loss

The last step is finding out how much mechanical energy was lost during the collision. This can be found by subtracting the final kinetic energy from the initial kinetic energy: \(\Delta KE = KE_{\text{initial}} - KE_{\text{final}}\). Performing the calculation gives the energy loss.

Key Concepts

Momentum

Momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. In practical terms, momentum is the product of an object's mass and velocity, expressed mathematically as \( p = m \cdot v \). Consider the example of a car and a truck involved in a collision. Before the crash, both the car and the truck have their respective momenta based on their mass and velocity:

• For the car: 1200 kg mass moving at 25.0 m/s, resulting in a momentum of 30000 kg·m/s.
• For the truck: 9000 kg mass moving at 20.0 m/s, resulting in a momentum of 180000 kg·m/s.

The total system's momentum is the sum of both individual momenta. According to the law of conservation of momentum, this total momentum must remain unchanged even after the collision. Thus, calculating the post-collision velocities involves applying this conservation principle, ensuring the total momentum before equals the total momentum after. This allows us to solve for unknown variables, like the truck's final velocity.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, mathematically defined as \( KE = 0.5 \cdot m \cdot v^2 \). The kinetic energy depends directly on the object's mass and the square of its velocity, emphasizing how a small change in speed results in a significant change in kinetic energy.

• For the car initially, the kinetic energy is calculated as \( 0.5 \cdot 1200 \, \text{kg} \cdot (25.0 \, \text{m/s})^2 \).
• The truck initially has a kinetic energy of \( 0.5 \cdot 9000 \, \text{kg} \cdot (20.0 \, \text{m/s})^2 \).

After the collision, both vehicles still possess kinetic energy, though distributed differently among them. The final kinetic energies reflect their new velocity configurations:

• The car at 18.0 m/s, providing a different kinetic energy calculation.
• The truck's new velocity, derived from conservation of momentum, alters its kinetic energy profile.

Ultimately, a comparison between initial and final kinetic energy helps in understanding energy distribution through the process of collisions.

Mechanical Energy Loss

Mechanical energy in the context of collisions often involves considering energy loss. When two objects collide, like our car and truck example, some initial mechanical energy turns into other forms, such as heat or sound due to deformation or friction.To find mechanical energy lost in collisions:

• Calculate the initial total kinetic energy before the collision by summing the kinetic energies of the moving bodies.
• Similarly, find the total kinetic energy after the collision, using above-derived velocities.

The energy loss is given by the difference: \( \Delta KE = KE_{\text{initial}} - KE_{\text{final}} \). This energy discrepancy manifests as the mechanical energy loss. It's crucial to remember that while momentum is conserved in collisions in isolated systems, kinetic energy isn't necessarily conserved, especially in inelastic collisions where objects may stick or deform. Evaluating this loss provides insight into physical dynamics of collision processes beyond mere motion, helping diagnose energy dispersion into less visible forms.