Two balls collide head-on on a table. The first ball has a mass of 0.50 kg and an initial velocity of 12.0 m/s. The second ball has a mass of 0.75 kg and an initial velocity of -16.0 m/s. After the collision, the first ball travels at a velocity of -21.6 m/s. What is the velocity of the second ball after the collision? Assume a perfectly elastic collision and no friction between the balls and the table
conserve momentum.
0.50(12.0) + 0.75(-16.0) = 0.50(-21.6) + 0.75v
To solve this problem, we can use the law of conservation of momentum and the law of conservation of kinetic energy.
The law of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be expressed as:
m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final
Where:
m1 and m2 are the masses of the first and second balls, respectively
v1_initial and v2_initial are the initial velocities of the first and second balls, respectively
v1_final and v2_final are the final velocities of the first and second balls, respectively
We are given:
m1 = 0.50 kg
v1_initial = 12.0 m/s
m2 = 0.75 kg
v2_initial = -16.0 m/s
v1_final = -21.6 m/s
Now, substituting these values into the equation, we get:
(0.50 kg) * (12.0 m/s) + (0.75 kg) * (-16.0 m/s) = (0.50 kg) * (-21.6 m/s) + (0.75 kg) * v2_final
Simplifying the equation further, we get:
6.0 kg·m/s - 12.0 kg·m/s = -10.8 kg·m/s + 0.75 kg · v2_final
Rearranging the equation, we can solve for v2_final:
0.75 kg · v2_final = 6.0 kg·m/s - 12.0 kg·m/s + 10.8 kg·m/s
0.75 kg · v2_final = 4.8 kg·m/s
Dividing both sides by 0.75 kg, we get:
v2_final = (4.8 kg·m/s) / (0.75 kg)
v2_final = 6.4 m/s
Therefore, the final velocity of the second ball after the collision is 6.4 m/s.
To find the velocity of the second ball after the collision, we can use the concepts of conservation of momentum and kinetic energy.
The law of conservation of momentum states that the total momentum of a system before a collision is equal to the total momentum after the collision.
In the given scenario, the initial momentum of the system is the sum of the momentums of the two balls before the collision:
Initial momentum = (mass of first ball * initial velocity of first ball) + (mass of second ball * initial velocity of second ball)
= (0.50 kg * 12.0 m/s) + (0.75 kg * -16.0 m/s)
= 6.0 kg m/s - 12.0 kg m/s
= -6.0 kg m/s
Using the law of conservation of momentum, the total momentum of the system after the collision will also be -6.0 kg m/s. Since the first ball has a velocity of -21.6 m/s after the collision, we need to find the velocity of the second ball.
Let's assume the velocity of the second ball after the collision is v.
The final momentum of the system will be given by:
Final momentum = (mass of first ball * final velocity of first ball) + (mass of second ball * final velocity of second ball)
Since we know the mass and velocity of the first ball after the collision, we can substitute the values:
-6.0 kg m/s = (0.50 kg * -21.6 m/s) + (0.75 kg * v)
Simplifying this equation, we have:
-6.0 kg m/s = -10.8 kg m/s + 0.75 kg * v
Rearranging the equation, we get:
0.75 kg * v = -6.0 kg m/s + 10.8 kg m/s
0.75 kg * v = 4.8 kg m/s
Dividing both sides of the equation by 0.75 kg, we find:
v = 4.8 kg m/s / 0.75 kg
v = 6.4 m/s
Therefore, the velocity of the second ball after the collision is +6.4 m/s.