Two billiard balls with identical masses and sliding in opposite directions have an elastic head-on collision. Before the collision, each ball has a speed of 20 cm/s. Find the speed of each billiard ball immediately after the collision.
momentum (total of zero) and energy (total of 2*(1/2) m v^2) are conserved. Therefore each rebounds at original speed but opposite direction.
Well, when it comes to billiard balls, things can get quite bouncy! In an elastic collision, the total momentum of the system is conserved. Since the masses and speeds of the balls are the same, we can assume they have equal magnitudes of momentum before the collision.
So, each ball has momentum p = m*v, where m is the mass and v is the velocity. Before the collision, both balls have a momentum of p = (mass)*(velocity) = (mass)*(20 cm/s).
Now, during the collision, their kinetic energy might change, but their total momentum remains the same. When two identical masses collide head-on elastically, they exchange speeds!
After the collision, the ball that was initially moving to the right has essentially "stolen" the speed of the initially left-moving ball, and vice versa. So, the speed of each ball immediately after the collision will still be 20 cm/s, but in the opposite direction.
In other words: both balls will leave the collision with the same magnitude of speed (20 cm/s) but with reversed directions.
I hope my answer didn't cushion the impact of the humor!
In an elastic collision, both the momentum and the kinetic energy of the system are conserved.
Let's assume the initial speed of both balls as v1 and v2, where v1 is the speed of ball 1 and v2 is the speed of ball 2.
Before the collision:
Ball 1 speed, v1 = 20 cm/s (given)
Ball 2 speed, v2 = -20 cm/s (opposite direction)
The negative sign for v2 indicates that the ball is moving in the opposite direction.
Using the principle of conservation of momentum:
Total initial momentum = Total final momentum
(mass of ball 1) * (initial velocity of ball 1) + (mass of ball 2) * (initial velocity of ball 2)
= (mass of ball 1) * (final velocity of ball 1) + (mass of ball 2) * (final velocity of ball 2)
Let's assume the masses of both balls as m1 and m2, where m1 = m2 (given).
(m1 * v1) + (m2 * v2) = (m1 * v1') + (m2 * v2')
After the collision:
The balls switch speeds:
Final speed of ball 1, v1' = -20 cm/s
Final speed of ball 2, v2' = 20 cm/s
Substituting the values back into the equation:
(m1 * v1) + (m2 * v2) = (m1 * v1') + (m2 * v2')
(m1 * 20 cm/s) + (m2 * (-20 cm/s)) = (m1 * (-20 cm/s)) + (m2 * 20 cm/s)
Simplifying the equation:
20 cm/s * (m1 - m2) = -20 cm/s * (m1 - m2)
The mass terms (m1 - m2) cancel out, as both balls have the same mass.
20 cm/s = -20 cm/s
This equation is true, but it does not provide meaningful information about the speeds of the balls after the collision.
Since the masses and initial velocities of the balls are the same, the velocity of each ball after the collision will remain the same as before the collision.
Therefore, the speed of each billiard ball immediately after the collision is 20 cm/s.
To find the speed of each billiard ball immediately after the collision, we can use the principle of conservation of momentum.
The momentum of an object is defined as the product of its mass and velocity. In an elastic collision, the total momentum of the system is conserved. Mathematically, we can express this as:
(m1 * v1) + (m2 * v2) = (m1 * v1') + (m2 * v2')
Where:
m1 and m2 are the masses of the two balls,
v1 and v2 are their initial velocities,
and v1' and v2' are their velocities immediately after the collision.
In this case, we are given that both balls have identical masses and speeds of 20 cm/s before the collision.
So, let's substitute the given values into the equation:
(1 * 20) + (1 * -20) = (1 * v1') + (1 * v2')
Simplifying the equation:
(20 - 20) = (v1' + v2')
Since both balls have the same mass, we can simplify further:
0 = v1' + v2'
This means that after the collision, the total momentum of the system will be zero. Therefore, the speeds of the balls after the collision must be equal in magnitude but opposite in direction.
Hence, the speed of each billiard ball immediately after the collision is 0 cm/s.