After a completely inelastic collision, two objects of the same mass and same initial speed are found to move away together at 1/5 their initial speed. Find the angle between the initial velocities of the objects.
The impulse given to a ball with mass of 4 kg is 16 N s. If the ball starts from rest, what is its final velocity?
To find the angle between the initial velocities of the objects after the collision, we first need to analyze the momentum conservation.
In an inelastic collision, the two objects stick together after the collision, forming a single object. Therefore, the final combined mass is the sum of the individual masses of the two objects.
Let's denote the initial velocities of the two objects as v1 and v2, and the final velocity of the combined object as vf.
According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be written as:
m1 * v1 + m2 * v2 = (m1 + m2) * vf
Since the masses of the two objects are the same, we can simplify the equation to:
v1 + v2 = 2 * vf
Now, we are given that after the collision, the objects move away together at 1/5 their initial speed. This means that the final velocity, vf, is equal to 1/5 times the initial velocity of either object. Mathematically, this can be written as:
vf = (1/5) * v1
Substituting this into the momentum conservation equation:
v1 + v2 = 2 * ((1/5) * v1)
v1 + v2 = (2/5) * v1
v2 = (2/5) * v1 - v1
v2 = (-3/5) * v1
Now, to find the angle between the initial velocities, we can use the trigonometric relationship between the components of the velocities. Let's denote the angle between the initial velocities as θ.
Since the objects have the same initial speed, the magnitudes of their velocities are equal. Thus, we can write:
|v1| = |v2|
From the equation for v2 above, we can see that v2 is a scalar multiple of v1. Therefore, the angle θ between the initial velocities is 180 degrees (or π radians).