Solving Collision Problem: Find Angle Between Initial Velocities

[jenavira]

Collision problem

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

I've got the equations
(2mv)cos theta1 = 2m.5v cos theta2
&
mv sin theta1 = 2m.5v sin theta2

But I'm not sure where to go from there. (Actually, I'm not sure how to get a numerical answer out of something I haven't been given any numbers to put into, so there's probably a conceptual thing I'm missing here...)

 

[Andrew Mason]

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

I've got the equations
(2mv)cos theta1 = 2m.5v cos theta2
&
mv sin theta1 = 2m.5v sin theta2

But I'm not sure where to go from there. (Actually, I'm not sure how to get a numerical answer out of something I haven't been given any numbers to put into, so there's probably a conceptual thing I'm missing here...)

Try using:

$$mvsin\theta + mvsin\alpha = 0$$

where $\theta$ and $\alpha$ are the respective angles of the initial velocities of the respective masses relative to the final velocity of the two masses together. This tells you that the angles are equal (after all, why should they be different?).

The rest should be obvious.

AM

 

[wais]

To solve this collision problem and find the angle between the initial velocities, we can use the conservation of momentum and conservation of kinetic energy equations. In an inelastic collision, the total momentum and total kinetic energy of the system are conserved.

Let's say the initial velocities of the two objects are v1 and v2, with an angle theta between them. After the collision, they move away together at a speed of 0.5v1 and an angle of 180 degrees. This means that the final velocity of the system is 0.5v1 at an angle of 180 degrees.

Using the conservation of momentum equation, we can write:

m1v1 + m2v2 = (m1 + m2)0.5v1

Simplifying this equation, we get:

v2 = (0.5m1v1)/(m2 + 0.5m1)

Similarly, using the conservation of kinetic energy equation, we can write:

0.5m1v1^2 + 0.5m2v2^2 = 0.5(m1 + m2)0.5v1^2

Simplifying this equation, we get:

v2 = (0.5m1v1)/(m2 + 0.5m1)

Now, we have two equations for v2, so we can equate them and solve for theta:

(0.5m1v1)/(m2 + 0.5m1) = (0.5m1v1)/(m2 + 0.5m1)

Solving for theta, we get:

theta = arccos(0.5)

Therefore, the angle between the initial velocities of the two objects is approximately 60 degrees. This means that the objects were initially moving towards each other at an angle of 60 degrees before the collision occurred.