Problem 45 - College Physics

Question

A \(1200-\mathrm{kg}\) car moving to the right with a speed of \(25 \mathrm{~m} / \mathrm{s}\) collides with a \(1500-\mathrm{kg}\) truck and locks bumpers with the truck. Calculate the velocity of the combination after the collision if the truck is initially (a) at rest, (b) moving to the right with a speed of \(20 \mathrm{~m} / \mathrm{s},\) and (c) moving to the left with a speed of \(20 \mathrm{~m} / \mathrm{s}\).

Step-by-Step Solution

Answer

(a) 11.11 m/s, (b) 22.22 m/s, (c) 0 m/s.

Step 1: Understand the Problem

We are given a collision problem where a car and a truck collide and stick together. We are to find the velocity of the combined mass after the collision for three different initial conditions of the truck. This is a conservation of momentum problem.

Step 2: Write the Conservation of Momentum Equation

The law of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. We can express this for the situation as:\[ m_1 u_1 + m_2 u_2 = (m_1 + m_2) v \]where:- \(m_1 = 1200\,\text{kg}\), \(u_1 = 25\,\text{m/s}\): mass and velocity of the car- \(m_2 = 1500\,\text{kg}\): mass of the truck- \(u_2\): initial velocity of the truck (varies for each scenario)- \(v\): final velocity of the combined system.

Step 3: Case A - Truck is at Rest

For this case, the initial velocity of the truck is \(u_2 = 0\,\text{m/s}\). Substitute these values into the conservation equation:\[ 1200 \times 25 + 1500 \times 0 = (1200 + 1500) v \]\[ 30000 = 2700v \]Solve for \(v\):\[ v = \frac{30000}{2700} \approx 11.11\,\text{m/s}\]

Step 4: Case B - Truck Moving to the Right

Now, the truck has an initial velocity of \(u_2 = 20\,\text{m/s}\). Substitute in the values:\[ 1200 \times 25 + 1500 \times 20 = (1200 + 1500) v \]\[ 30000 + 30000 = 2700v \]\[ 60000 = 2700v \]Solve for \(v\):\[ v = \frac{60000}{2700} \approx 22.22\,\text{m/s}\]

Step 5: Case C - Truck Moving to the Left

For this case, the truck's initial velocity is \(u_2 = -20\,\text{m/s}\) (negative because it's to the left). Substitute into the equation:\[ 1200 \times 25 + 1500 \times (-20) = (1200 + 1500) v \]\[ 30000 - 30000 = 2700v \]\[ 0 = 2700v \]So, \( v = 0\,\text{m/s}\), which means the system is at rest after the collision.

Key Concepts

Collision Problem

In physics, a collision problem involves two or more interacting bodies that come in contact with each other, exchanging momentum and energy. A characteristic feature of collision problems is that they often require the use of the conservation of momentum to solve them. In this particular exercise, we're presented with a car and a truck that collide and end up moving together after the impact. Such collisions, where objects stick together post-collision, are known as perfectly inelastic collisions.

These types of problems are interesting because they allow you to explore different possible outcomes based on varying initial conditions. For example, the truck's initial velocity impacts the final speed of the combined car and truck system. To solve collision problems, it's crucial to comprehend the relationship between the individual objects involved and how their momenta interact upon collision.

Momentum Equation

The momentum equation is a fundamental concept in physics that expresses the product of an object's mass and its velocity. It is symbolically represented as \( p = m imes u \). This equation is crucial for the conservation of momentum principle, which states that in an isolated system, the total momentum before and after a collision remains constant.

For this exercise, we formulate an equation based on this principle:

• Before the collision: \( m_1 u_1 + m_2 u_2 \)
• After the collision: \( (m_1 + m_2) v \)

Consequently, setting these expressions equal to each other helps determine the final velocity of the combined mass. This equation allows us to calculate the results under different scenarios, emphasizing that the total system momentum is preserved no matter the initial conditions, as long as external forces are absent.

Velocity Calculation

Velocity calculation in collision problems is about finding the system's speed after two objects collide and combine their masses. To compute this velocity, we apply the conservation of momentum equation derived from the masses and initial velocities of the colliding entities.

In our context, the car and the truck lock together post-collision. When calculating the final velocity of such a combined system, adjust the initial condition for each case:

• Case A: The truck is at rest (\( u_2 = 0 \)).
• Case B: The truck moves to the right (\( u_2 = 20 \, \text{m/s} \)).
• Case C: The truck moves to the left (\( u_2 = -20 \, \text{m/s} \)).

For each situation, simply substitute the known values into the momentum equation, solve for \( v \), and smoothly derive the final velocity. Understanding how different initial velocities affect the calculation helps reinforce the practical application of the momentum equation.

Physics Problem Solving

Physics problem solving involves breaking down complex scenarios into understandable parts and systematically applying relevant principles to find a solution. When approaching a collision problem, like the one presented, follow these steps to effectively solve it:

• Understand the scenario: Identify all bodies involved, their masses, and initial velocities.
• Choose the right principle: Use conservation laws, here the conservation of momentum, to determine the required physical quantities.
• Execute the solution: Formulate equations based on the chosen principle and solve them step by step for each condition.
• Analyze results: Compare the outcomes of different scenarios to gain insights into the problem.

By adhering to a structured approach, you enhance your comprehension and can tackle similar collision problems in physics robustly. Each step brings clarity and accuracy, ensuring you solve physics problems effectively.