91Ó°ÊÓ

In reconstructing an automobile accident, investigators study the total momentum, both before and after the accident, of the vehicles involved. The total momentum of two vehicles moving in the same direction is found by multiplying the weight of each vehicle by its speed and then adding the results. \({ }^{19}\) For example, if one vehicle weighs 3000 pounds and is traveling at 35 miles per hour, and another weighs 2500 pounds and is traveling at 45 miles per hour in the same direction, then the total momentum is \(3000 \times 35+2500 \times 45=\) 217,500 . In this exercise we study a collision in which a vehicle weighing 3000 pounds ran into the rear of a vehicle weighing 2000 pounds. a. After the collision, the larger vehicle was traveling at 30 miles per hour, and the smaller vehicle was traveling at 45 miles per hour. Find the total momentum of the vehicles after the collision. b. The smaller vehicle was traveling at 30 miles per hour before the collision, but the speed \(V\), in miles per hour, of the larger vehicle before the collision is unknown. Find a formula expressing the total momentum of the vehicles before the collision as a function of \(V\). c. The principle of conservation of momentum states that the total momentum before the collision equals the total momentum after the collision. Using this principle with parts a and b, determine at what speed the larger vehicle was traveling before the collision.

Step by step solution

01

Calculate Post-Collision Momentum

After the collision, the larger vehicle weighs 3000 pounds and travels at 30 mph, while the smaller vehicle weighs 2000 pounds and travels at 45 mph. The total momentum post-collision is calculated as follows:\[\text{Total Momentum} = (3000 \times 30) + (2000 \times 45)\]Computing the products:\[3000 \times 30 = 90,000\]\[2000 \times 45 = 90,000\]Adding these results gives us:\[90,000 + 90,000 = 180,000\]

02

Formulate Pre-Collision Momentum Expression

Before the collision, the smaller vehicle weighs 2000 pounds and was traveling at 30 mph, while the larger vehicle's speed is unknown and represented by \( V \). Express the total pre-collision momentum as a function of \( V \):\[\text{Pre-collision Momentum} = (3000 \times V) + (2000 \times 30)\]Calculate the known part:\[2000 \times 30 = 60,000\]Thus, the expression for total pre-collision momentum is:\[3000V + 60,000\]

03

Apply Conservation of Momentum

According to the principle of conservation of momentum, the total momentum before the collision equals the total momentum after the collision. Set the pre-collision momentum equal to the post-collision momentum:\[3000V + 60,000 = 180,000\]

04

Solve for the Unknown Speed \( V \)

Begin by isolating \( V \) in the equation:\[3000V = 180,000 - 60,000\]Simplify the right side:\[3000V = 120,000\]Now, divide both sides by 3000 to solve for \( V \):\[V = \frac{120,000}{3000} = 40\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Momentum Calculation

Momentum is a key concept in physics, crucial for understanding how objects behave during motion, especially in collisions. In simple terms, momentum is the product of an object's mass and velocity.
When calculating momentum, particularly for moving vehicles, one follows a straightforward formula:

• Momentum = Mass \( \times \) Velocity

In the case provided in the exercise, to find the momentum of each vehicle after the collision, we use this formula. The larger vehicle, weighing 3000 pounds and moving at 30 miles per hour, has a momentum of \(3000 \times 30 = 90,000\).
Similarly, the smaller vehicle weighing 2000 pounds and traveling at 45 miles per hour has a momentum of \(2000 \times 45 = 90,000\).
By summing these individual momenta, we find the total momentum after the collision:

• Total Momentum = 90,000 + 90,000 = 180,000

Collision Analysis

Collision analysis is a method used to understand the dynamics of objects before and after they collide. This involves looking at the momentum of each object involved in the collision.
The key principle here is the conservation of momentum, which states that the total momentum of a closed system remains constant if no external forces act on it.
In the case of the collision described, we calculate the total momentum before and after the incident.
Before the collision, the momentum of the smaller vehicle traveling at 30 mph with a weight of 2000 pounds is \(2000 \times 30 = 60,000\).
For the larger vehicle, since its initial speed is unknown, we express its momentum in terms of this unknown speed \(V\):

• Total Pre-collision Momentum = \(3000V + 60,000\)

Unknown Variable Determination

Determining unknown variables is an essential aspect of collision analysis. In this scenario, we want to find the speed of the larger vehicle before the collision, which is represented as \(V\).
Using the principle of conservation of momentum, we set the total pre-collision momentum equal to the total post-collision momentum, as momentum is conserved in elastic collisions:

• \(3000V + 60,000 = 180,000\)

We solve for the unknown speed \(V\) by isolating it on one side of the equation. First, subtract 60,000 from both sides:

• \(3000V = 120,000\)

Then, divide both sides by the mass of the larger vehicle (3000) to solve for \(V\):

• \(V = \frac{120,000}{3000} = 40\)

This calculation reveals that the larger vehicle was traveling at 40 miles per hour before the collision. By breaking down steps, we can understand how to determine unknowns using fundamental principles of physics.