After a completely inelastic collision two objects of the same mass and the same initial speed are found to move away together at 1/2 thier initial speed. Find the angle bewteen the initial velocitie of the objects?
Can you please show me the step to solve this problem? I am really confused. I think I would use the conservation of momentum but can you show me how to solve this? Thanks for the help.
From the conservation of momentum, the angles are equal from the direction of the moving mass after reaction. Now, then in the direction of the final direction...
2mcosTheta*v= (m+m)v/2
solve for theta.
To solve this problem, you are correct in using the conservation of momentum. Here are the steps to solve it:
Step 1: Write down the initial momentum of the system before the collision. Since both objects have the same mass (m) and the same initial speed (v), their momenta are equal and opposite in direction. Therefore, the initial momentum is:
P_initial = -m*v - m*v
Step 2: Write down the final momentum of the system after the collision. Since the objects move away together at 1/2 their initial speed, their final momentum is:
P_final = (m+m)*(1/2)*v
Step 3: Apply the conservation of momentum, which states that the initial momentum is equal to the final momentum:
P_initial = P_final
Substituting the values from steps 1 and 2:
-m*v - m*v = (m+m)*(1/2)*v
Simplifying the equation:
-2m*v = 2m*(1/2)*v
-2m*v = m*v
Dividing both sides by -2m:
v = -v/2
Step 4: Solve for the angle theta by equating the x-components of the final velocities of the two objects.
In the x-direction, the initial velocities of both objects are equal and opposite, so their resultant x-component is zero. The final x-component of the velocity is given by:
v_x = v*cos(theta)
Since the objects move away together in the final direction, their x-component of velocity is:
(v/2)*cos(theta)
Equating the x-components:
v*cos(theta) = (v/2)*cos(theta)
Cancelling out v:
cos(theta) = (1/2)*cos(theta)
Step 5: Solve for theta:
cos(theta) = (1/2)*cos(theta)
Multiply both sides by 2:
2*cos(theta) = cos(theta)
Subtract cos(theta) from both sides:
cos(theta) = 0
Since cosine(theta) equals zero, the angle theta must be 90 degrees or 270 degrees.
Therefore, the angle between the initial velocities of the objects is either 90 degrees or 270 degrees.