Question

After a totally inelastic collision, two objects of the same mass and initial speed are found to move away together at half their initial speed. Find the angle between the initial velocities of the objects.

Short answer

The angle between the initial velocities of the objects is 0 degrees.

Step-by-step solution

Understanding the problem

The objects of the same mass stick together after undergoing a totally inelastic collision and move with half their initial speed. Let's denote the initial speed as \(v\), and the masses of the objects as \(m\). The speed after the collision is \(v/2\), and the total mass after the collision is \(2m\). From the conservation of momentum, we get \(2mv = 2m \times v/2\).

Setting up the equation

The previous equation simplifies to \(v = v/2\), which does not seem to help at all but in fact it tells us that the two initial velocities are in a perfect angle so the result of their vector addition (the common final velocity) has half the magnitude of each initial one. The x-component of 1 of the velocities cancels the x-component of the other leaving a final y-component that, due to having half magnitude than the initial for both, is half the y-component of each one.

Calculating the required angle

From this discussion it can be deduced that the angle we are interested in would be simply 2 times the angle whose cosine is the result of multiplying the magnitude of the final velocity (1/2v) by 2 and dividing by the magnitude of one fo the initial velocities (v), so we get \(2\times \arccos((2v/2)/v) = 2\times \arccos(1) = 2\times 0\).

Key concepts

Totally Inelastic Collision

When we talk about a totally inelastic collision, we are referring to a type of collision where the colliding objects stick together afterward. This is different from an elastic collision, where the objects bounce off each other and kinetic energy is conserved. In a totally inelastic collision, the maximum amount of kinetic energy is lost, but crucially, momentum is still conserved. This is because the law of momentum conservation is universally applicable for all types of collisions in a closed system regardless of whether they are elastic or inelastic.

A good real-life example of a totally inelastic collision is when two cars crash and move together as a single wreckage. The fact that the colliding objects coalesce means their post-collision motion can be treated as a single entity. This concept of sticking together is vital for solving problems involving inelastic collisions, as it simplifies the final state of the system into one where a single object with combined mass moves with a common velocity.

Conservation of Momentum

The principle of conservation of momentum states that the total momentum of a closed system remains constant if no external forces are acting upon it. Momentum can be understood as the 'quantity of motion' an object has, and it's calculated as the product of an object’s mass and velocity (momentum = mass x velocity).

When we apply this principle to the problem at hand, we see that the sum of the momenta of both objects before the collision must be equal to their combined momentum after the collision. Since the objects have the same mass and initial speed, but are traveling at an angle to one another before they collide, the total momentum is a vector that consists of the combination of each object's momentum. Using vector addition and some geometry, we can find the angle because the momentum before and after the collision is conserved even though their speeds (and therefore kinetic energies) change.

Vector Addition

Moving onward, vector addition is how we combine the velocities or momenta of the two objects in our inelastic collision problem. Since velocity and momentum are vector quantities (meaning they have both magnitude and direction), they must be combined according to the rules of vector addition. When two vectors are added, the resultant vector is found by placing the tail of the second vector to the head of the first one, forming a triangle, and then drawing a vector from the tail of the first vector to the head of the second one.

This graphical method helps visualize the concept, but mathematically, we often break vectors down into their x and y components and add those separately. In our problem, the complete cancelation of the x-components and the maintenance of the y-components at half value is due to such vector addition, which is essential for determining the outcome of the collision in terms of both speed and direction.

Calculating Angles in Collisions

Finally, the challenge of calculating angles in collisions generally requires a bit of trigonometry. In the specific scenario of our inelastic collision problem, it involves finding the angle between the initial velocity vectors of the two objects. Since these angles affect the resultant velocity vector, we need to understand how to use trigonometric functions, such as cosine and sine, as well as their inverses, to solve for these angles based on the components of the vectors.

In our case, the solution hinges upon understanding how the direction components of the velocity vectors interact through the collision. We do this by examining the relationship between the final combined velocity vector and what we know about the initial velocities and their respective directional components. The perfect cancelation of the x-components in the vector addition process leads us to determine the final angle using trigonometric principles, specifically the cosine function and its inverse, arccos.