Find the final velocity of the two balls if the ball with velocity v2i = -19.5 cm/s has a mass equal to half that of the ball with initial velocity v1i = +26.8 cm/s.


PLEASE HELP!! i have tried so many different ways and can't seem to get the answer

To find the final velocity of the two balls, we can use the principle of conservation of linear momentum. According to this principle, the total linear momentum before the collision is equal to the total linear momentum after the collision.

Let's assume the ball with initial velocity v1i = +26.8 cm/s has mass m1 and the ball with initial velocity v2i = -19.5 cm/s has mass m2.

The linear momentum (p) of an object is given by the product of its mass (m) and velocity (v), i.e., p = m * v.

Before the collision, the total linear momentum is given by the sum of the individual momenta of the two balls:

p1i + p2i = m1 * v1i + m2 * v2i

After the collision, the total linear momentum is given by the sum of the individual momenta of the two balls:

p1f + p2f = m1 * v1f + m2 * v2f

According to the conservation of linear momentum:

p1i + p2i = p1f + p2f

Substituting the given initial velocities:

m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f

Now, let's substitute the given values to calculate the final velocity.

Given:
v1i = +26.8 cm/s
v2i = -19.5 cm/s
m2 = (1/2) * m1

Substituting these values into the equation:

m1 * (+26.8 cm/s) + [(1/2) * m1] * (-19.5 cm/s) = m1 * v1f + [(1/2) * m1] * v2f

Simplifying further:

26.8 m1 - 9.75 m1 = m1 * v1f + 0.5 m1 * v2f

17.05 m1 = m1 * v1f + 0.5 m1 * v2f

Now, we can divide both sides of the equation by m1 to eliminate the mass factor:

17.05 = v1f + 0.5 * v2f

Finally, rearranging the equation to solve for the final velocity of the balls:

v1f + 0.5 * v2f = 17.05

v1f = 17.05 - 0.5 * v2f

Now, here's how to calculate the final velocity:

1. Substitute the given value of v2i into the equation:
v2i = -19.5 cm/s

2. Substitute the given value of v1i into the equation:
v1i = +26.8 cm/s

3. Substitute the mass relationship m2 = (1/2) * m1 into the equation.

4. Substitute the calculated values of v2i, v1i, and m2 into the equation:
v1f + 0.5 * v2f = 17.05

5. Rearrange the equation to solve for v1f:
v1f = 17.05 - 0.5 * v2f

6. Calculate the final velocity of the second ball by solving the equation.

By following these steps, you should be able to calculate the final velocity of both balls.