Two particles with the same mass and same initial speed collide and stick together and leave the collision with half of their initial speed, bound together as one particle. What was the angle between the initial momenta of the particles before the collision?

To find the angle between the initial momenta of the particles before the collision, we can use the principle of conservation of momentum.

Let's assume that the initial momenta of the two particles before the collision are represented by vectors 𝑝1 and 𝑝2. Since the particles have the same mass and same initial speed, their initial momenta will have the same magnitude, but may have different directions.

After the collision, the two particles stick together and move as one particle. Since they leave the collision with half of their initial speed, the magnitude of their final momentum will be half of the magnitude of their initial momentum.

According to the principle of conservation of momentum, the total momentum before the collision should be equal to the total momentum after the collision. Mathematically, this can be expressed as:

𝑝1 + 𝑝2 = 𝑝

Where 𝑝 represents the momentum of the combined particles after the collision.

Since the magnitudes of 𝑝1 and 𝑝2 are equal, and the magnitude of 𝑝 is half of 𝑝1 or 𝑝2, we can express the above equation as:

2𝑝1cos(θ) = 0.5𝑝1

where 𝜃 represents the angle between the initial momenta 𝑝1 and 𝑝2.

Simplifying the equation:

2cos(θ) = 0.5

Dividing both sides by 2:

cos(θ) = 0.25

To find the angle 𝜃, we can take the inverse cosine (arccos) of 0.25:

𝜃 = arccos(0.25)

Using a calculator or math software, we find:

𝜃 ≈ 75.52 degrees

Therefore, the angle between the initial momenta of the particles before the collision is approximately 75.52 degrees.