# 2-D Linear Momentum, Collision Problem

• omicronhuh
In summary, the conversation discusses a scenario where the listener is working as an intern for the Montreal Police Collision Investigation unit. They are tasked with assisting investigators in determining the circumstances surrounding collisions and taking action against drivers at fault. The listener is given a specific collision to investigate and is asked to take photos and measurements at the scene. They also have to calculate the initial velocities and determine the cause of the collision using momentum and energy conservation equations. The conversation ends with the listener seeking advice on how to proceed with their calculations.
omicronhuh
Good evening! I'm having trouble completing an assignment related to inelastic collisions and momentum.

## Homework Statement

Here is the statement:

You are in great shape on this hot July morning. You are working as an intern for the Montreal Police (SPVM) Collision Investigation unit. You assist the investigators in order to determine, as accurately as possible, the circumstances surrounding collisions (initial velocity of the vehicles involved, direction vehicles were heading, etc.) in order to take action against drivers at fault under the Criminal Code (dangerous driving causing death or other charges).

You just get to the scene of a collision involving two vehicles. The Chief Investigating Officer addresses you, “Come here quick, we need you. You know the routine, so get to work.” Your job is to take photos of the vehicles as they are positioned, and take a certain number of measurements, including the length of the skid marks and layout of the debris. You immediately draw the following sketch of the accident and include a few common measurements.Your findings are as follows:
• Numerous debris (broken glass, plastic, etc.) were found at a distance of 12.63 m from the red vehicle
• (2) and 11.87 m from the yellow vehicle (1);
• The yellow vehicle (1) seems to have braked in a straight line;
• The red vehicle (2) veered 6.5º away from its initial course;
• The marks of the impact on the red vehicle (2) show that it was heading 98º away from the yellow
• vehicle (1) at the moment of collision;
• There are skid marks over a distance of 30 m before the debris;
• The posted speed limit on this street is 70 km/h;
• Mass of vehicle 1 (yellow) 2674 kg, mass of vehicle 2 (red) 1110 kg.

I also found out that the friction coefficient of the pavement is mk = 0.86

I'll include the sketch as well:

I'm using the following velocity indicators and ignoring the ones in the sketch.

v1i = initial velocity of the yellow car
v2i = initial velocity of the red car
v1c = velocity of the yellow car just after the collision
v2c = velocity of the red car just after the collision

## Homework Equations

This are the relevant equations:
p=mv
m1v1 + m2v2 = (m1 + m2)vf
F = ma
And conservation of energy equations: U1 + K1 + Q11 = U2 + K2 + Q12

## The Attempt at a Solution

My attempts at solving this problem are as follows:
I'm starting at point c and comparing it to b to then compare b and a using momentum to solve the problem.

In my first attempt I used energy conservation to try to calculate the velocity of the cars just after the collision.
So K = Q, in this case.

1/2(m1)(v1^c)^2 + 1/2(m2)(v2c)^2 = mk(N1)(L) + mk(N2)(L)
1337v1c^2 + 555v2c^2 = 386,056.11

And I hit a roadblock, I couldn't figure out what other equations to use to get the velocity of either car right after the collision.

So, I calculated the deceleration the cars suffered thanks to the friction. I multiplied N for the friction coefficient and then divided by the mass.

a = -8.43m/s^2

Then with kinematics equations I got the velocity of the cars right after the impact.
v1c^2=-2a(x-x0)
v1c^2 = -2(-8.43)(11.87)
v1c = 14.14 m/s

v2c^2 in the x axis=-2a(x-x0)(cos 6.5)
v2c^2 = -2(-8.43)(12.63)(cos 6.5)
v2c in the x-axis = 14.54 m/s

v2c^2 in the x axis=-2a(x-x0)(sin 6.5)
v2c^2 = -2(-8.43)(12.63)(sin 6.5)
v2c in the y-axis = 4.9 m/s

And with this I try to use the momentum conservation equation:
m1v1i + m2v2i (cos 98) = m1(14.14) + m2(14.54)
2674v1i + 1110(-0.139)v2i = 53,949.76

And is at this point where I'm not sure how to proceed. I've tried using the same deceleration to know v1i with kinematics equations and then using that to know v2i, but the speed that results for the red car turns out to be quite absurd, nearing the 400km/h. I am also not sure I'm using the angles correctly.

Do you guys have any advice for me?
Thanks!

Looks to me that the vehicles separate (just) after the collision. The red car goes further. So you have two separate energy equations for their subsequent skids. But you effectively did that when you switched to using the kinematics equations.
Not sure how you used kinematics to find "v1i". Are you confusing two different velocities for car 1, the velocity before it started braking and tne velocity just before impact?

You have not used momentum conservation in the y direction during impact.

Last edited:

## 1. What is 2-D linear momentum?

2-D linear momentum is a physical quantity that describes the motion of an object in two dimensions. It is the product of an object's mass and its velocity in the x and y directions.

## 2. How is 2-D linear momentum calculated?

2-D linear momentum is calculated by multiplying an object's mass by its velocity in the x and y directions. The equation for 2-D linear momentum is p = mv, where p is momentum, m is mass, and v is velocity.

## 3. What is the conservation of 2-D linear momentum?

The conservation of 2-D linear momentum states that in a closed system, the total momentum in the x and y directions remains constant before and after a collision or interaction between objects. This means that the total momentum in the x and y directions must be the same before and after the collision.

## 4. How do you solve a collision problem using 2-D linear momentum?

To solve a collision problem using 2-D linear momentum, you must first calculate the initial and final momentum in the x and y directions for each object involved in the collision. Then, set the initial and final momentums equal to each other and solve for the unknown variables.

## 5. What are some real-life applications of 2-D linear momentum?

2-D linear momentum is used in many real-life applications, such as calculating the trajectory of a projectile, predicting the movement of objects in a game of billiards, and understanding the forces involved in a car crash. It is also important in engineering and designing structures to withstand forces and in analyzing the motion of celestial bodies.

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