Problem 13 - Applied Physics

Question

A \(2450-\mathrm{kg}\) automobile moving north \(12.0 \mathrm{~m} / \mathrm{s}\) collides with a \(1650-\mathrm{kg}\) automobile moving \(8.00 \mathrm{~m} / \mathrm{s}\) on an icy road. The automobiles stick together and move after the collision. Find the velocity of the automobiles after the collision if the automobiles were traveling in (a) opposite directions and (b) the same direction before the collision.

Step-by-Step Solution

Answer

(a) 3.95 m/s north; (b) 10.39 m/s north.

Step 1: Understand the Concept

This problem involves conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision because it is a closed system and there are no external forces acting.

Step 2: Identify Known Variables

We have:- Mass of first car, \( m_1 = 2450 \) kg,- Velocity of first car, \( v_1 = 12.0 \) m/s,- Mass of second car, \( m_2 = 1650 \) kg,- Velocity of second car, \( v_2 = 8.00 \) m/s.The cars stick together after the collision, so we will find their combined velocity.

Step 3: Setup the Conservation of Momentum Equation (Opposite Directions)

Before the collision, the total momentum in the north direction is given by:\[ m_1 v_1 - m_2 v_2 \]After the collision, the momentum is:\[ (m_1 + m_2) v_f \]Thus,\[ m_1 v_1 - m_2 v_2 = (m_1 + m_2) v_f \]

Step 4: Calculate the Velocity After Collision (Opposite Directions)

Substituting the numbers:\[ 2450 imes 12.0 - 1650 imes 8.00 = (2450 + 1650) v_f \]\[ 29400 - 13200 = 4100 v_f \]\[ 16200 = 4100 v_f \]\[ v_f = \frac{16200}{4100} \approx 3.95 \text{ m/s (north)} \]

Step 5: Setup the Conservation of Momentum Equation (Same Direction)

Before the collision, the total momentum in the north direction when both cars move in the same direction is given by:\[ m_1 v_1 + m_2 v_2 \]After the collision, the momentum is:\[ (m_1 + m_2) v_f \]Thus,\[ m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f \]

Step 6: Calculate the Velocity After Collision (Same Direction)

Substituting the numbers:\[ 2450 imes 12.0 + 1650 imes 8.00 = (2450 + 1650) v_f \]\[ 29400 + 13200 = 4100 v_f \]\[ 42600 = 4100 v_f \]\[ v_f = \frac{42600}{4100} \approx 10.39 \text{ m/s (north)} \]

Key Concepts

Collision Physics

Understanding collision physics is essential for solving problems involving moving objects. A collision occurs when two or more objects hit each other. This interaction can happen in various environments, like icy roads in this scenario.

Collisions are categorized based on whether the objects stick together or bounce apart after impact:

• Inelastic Collisions: The objects stick together, as seen in the problem.
• Elastic Collisions: The objects bounce off each other.

Collision physics helps us analyze and predict the motion of objects post-impact. This involves using laws like the conservation of momentum, which states that the total momentum of a closed system remains constant if there are no external forces acting.

Momentum Calculation

Calculating momentum is crucial in solving collision problems. Momentum, a product of an object's mass and velocity, measures how hard it is to stop the object. It is given by the formula: \[ p = mv \]where:

• \( p \) is momentum,
• \( m \) is mass,
• \( v \) is velocity.

In the problem, we calculated the momentum of each automobile before the collision. For momentum in opposite directions, we subtract the momentum of the second car from the first. If they move in the same direction, we add their momenta. This calculation lays the groundwork for applying the conservation of momentum when determining the velocities post-collision.

Physics Problem Solving

Solving physics problems like this requires a methodical approach. Start by identifying the type of collision and the relevant laws, such as the conservation of momentum, applicable here.

Next, gather all known variables, such as masses and velocities, and use them to set up relevant equations. In this exercise:

• For opposite directions, the equation is: \( m_1v_1 - m_2v_2 = (m_1 + m_2)v_f \)
• For the same direction, use: \( m_1v_1 + m_2v_2 = (m_1 + m_2)v_f \)

Carefully substitute the known values and solve the equations to find the unknowns. Breaking problems into manageable steps helps avoid mistakes and makes complex concepts more approachable.

Inelastic Collision

An inelastic collision is where colliding objects stick together after impact, moving with a common velocity. Energy is not conserved in terms of kinetic energy, but momentum is always conserved. In this exercise, after the collision, the velocity can be found by applying the principle of the conservation of momentum. The resultant equations for both scenarios (opposite and same direction) reflect the combined mass of the vehicles moving with the final velocity. Though the cars may lose speed due to deformation and energy transfer, understanding that momentum remains conserved is key to solving the problem. This exercise demonstrates practical use of physics in real-life situations, helping clarify this fundamental concept.