At the interscetion of Texas Avenue and University Drive, a yellow subcompact car with mass 950

At the interscetion of Texas Avenue and University Drive, a...

Key Concepts

Conservation of Linear Momentum

In isolated systems, the total linear momentum is conserved during collisions. This means that the vector sum of the momenta of all objects before the collision is equal to the vector sum after the collision. This principle applies separately in each independent direction, typically decomposing the momentum into its horizontal and vertical components for analysis.

Inelastic Collision

An inelastic collision is one in which the colliding objects stick together after impact, resulting in a loss of kinetic energy even though momentum is conserved. Analyzing such collisions involves applying the conservation of momentum in each direction and acknowledging that the kinetic energy before and after the collision will not be the same.

Vector Decomposition

When dealing with collisions in two dimensions, it is essential to decompose velocities and momenta into their components along mutually perpendicular axes. This breakdown simplifies the application of conservation laws by allowing the calculations to be performed in each independent direction, typically using trigonometric functions to resolve angles.

Transcript

00:01 In this question, one car with a mass of 950 kilograms was moving to the east, or in this case to the right.

00:08 And another car with a mass of 1 ,900 kilograms was moving to the north.

00:14 The velocity of both vehicles was unknown.

00:17 Then they collide somewhere here, and after that collision, both vehicles traveled together, with a velocity of 16 meters per second in the direction, making 24 degrees.

00:30 East of the north.

00:32 Then we have to determine what was the initial velocity of both cars.

00:37 For that we have to use the law of conservation of momentum.

00:41 That law tells us that the net momentum is conserved in every situation.

00:47 Therefore it tells us that the net momentum before the collision is equal to the net momentum after the collision.

00:56 But then notice that in this question we are working with two different directions, one direction which is the vertical one and another direction which is the horizontal one.

01:07 So we must be careful and remember that this is a vectorial law.

01:12 That is, the momentum is a vector and it can point in any direction.

01:17 Okay, that law must hold in both directions separately.

01:22 That is, this is a vector law, so it must be true for the x -axis, which is the horizontal axis that i'm choosing like this, and for the y -axis, which is the vertical axis that i'm choosing like this.

01:36 So that law can be written as follows.

01:39 For the x -axis, it's true that the net momentum in the x -direction before the collision is equal to the net momentum in the x direction after the collision.

01:51 In the y -direction, it's true that the net momentum, in the y -direction before the collision is equal to the net momentum in the y direction after the collision.

02:02 So, using these equations, we will be able to determine what was the velocities of cars number one and two.

02:09 So, it goes as follows.

02:11 Let us work before with the x direction.

02:15 So, in the x direction, the net momentum before the collision was composed only by the momentum of car number one, because it's the only car that has some velocity in that direction.

02:27 So, the net momentum before the collision is given by the mass of car number 1 m1 times the velocity of car number 1, which is v1.

02:39 And then after the collision, what happens is that both cars are moving in this direction with a velocity of 16 meters per second.

02:48 So as you can see, this velocity has a component in the x direction.

02:52 That is, we can draw that velocity like that.

02:57 And by doing that, you can decompose this velocity in its two components, and then it's very easy to see that it points both in the y direction and in the x direction.

03:08 So, the momentum after the collision is given by the sum of the masses, so the total mass, m1 plus m2, times the velocity after the collision.

03:20 Let me call this big v.

03:21 In the x direction, of course.

03:24 Then, the velocity of car 1 before the collision is equal to the mass of car number 1 plus the mass of car number 2 times the velocity after the collision divided by the mass of car number 1.

03:38 And this can be written as follows.

03:40 V1 is equal to 1 plus m2 divided by m1 times the velocity after the collision x component.

03:49 Now let us plug in the values in this equation.

03:53 V1 is equal to 1 plus the mass of car number 2, 1 ,900, divided by the mass of car number 1, 950 times the x component of the velocity after the collision, that is vx.

04:08 How can we calculate vx given v.

04:11 It's not difficult to do.

04:13 You just have to notice that we can make a triangle out of the velocity and its components.

04:18 The triangle is just like this...

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