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.
To find the final velocity of the two balls, we can use the law of conservation of momentum, which states that the total momentum before a collision is equal to the total momentum after the collision, assuming there are no external forces acting on the system.
The formula for momentum is given by:
p = m * v
where p is the momentum, m is the mass, and v is the velocity.
Since we know the initial velocity of the two balls, we can calculate the initial momentum for each ball.
For Ball 1:
m1 = mass of Ball 1 = m
v1i = initial velocity of Ball 1 = +26.8 cm/s
So, the initial momentum of Ball 1, p1i, is given by:
p1i = m1 * v1i
For Ball 2:
m2 = mass of Ball 2 = (1/2) * m (as given)
v2i = initial velocity of Ball 2 = -19.5 cm/s
So, the initial momentum of Ball 2, p2i, is given by:
p2i = m2 * v2i
Now, since momentum is conserved, the total initial momentum of the system (p_total) is equal to the total final momentum of the system (p_total):
p_total = p1i + p2i = p1f + p2f
Since the final momentum depends on the final velocities of the two balls, let's call them v1f and v2f:
p1i + p2i = m1 * v1f + m2 * v2f
We can substitute the values we know into this equation:
m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f
Now, we can solve for v1f and v2f. Let's plug in the values we have:
m * (+26.8 cm/s) + [(1/2) * m] * (-19.5 cm/s) = m * v1f + [(1/2) * m] * v2f
Simplifying this equation:
26.8 cm/s - (19.5/2) cm/s = v1f + (1/2) * v2f
Now, we have an equation that relates the final velocities of the two balls. We can further simplify this equation by substituting the given relationship between the masses:
26.8 cm/s - (19.5/2) cm/s = v1f + (1/2) * v2f
Solving this equation will give us the value for the final velocity of Ball 1 (v1f) and the final velocity of Ball 2 (v2f) relative to their initial velocities (v1i and v2i).