Two balls have masses 44 kg and 86 kg. The 44 kg ball has an initial velocity of 80 m/s (to the right) along a line joining the two balls and the 86 kg ball is at rest. The 86 kg ball has initial velocity of −24 m/s. The two balls make a head-on elastic collision with each other.
(1)What is the final velocity of the 44 kg ball?
(2)The final velocity of the 86Kg ball?
Answer in units of m/s
you have to use both conservation of momentum and energy.
Start with conservation of momentum, solve for one of the ball's velocity in terms of all the other variables, then substitute that into the conservation of energy.
The algebra is substantial, so be prepared.
To solve this problem, we can use the principle of conservation of momentum and the equation for elastic collisions. The momentum before the collision is equal to the momentum after the collision. We can write this as:
m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f
where m1 and m2 are the masses of the balls, v1i and v2i are the initial velocities, and v1f and v2f are the final velocities.
Given:
m1 = 44 kg
v1i = 80 m/s (to the right)
m2 = 86 kg
v2i = -24 m/s
(1) To find the final velocity of the 44 kg ball, v1f, we can rearrange the equation:
m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f
v1f = (m1 * v1i + m2 * v2i - m2 * v2f) / m1
(2) To find the final velocity of the 86 kg ball, v2f, we can rearrange the equation:
m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f
v2f = (m1 * v1i + m2 * v2i - m1 * v1f) / m2
Now let's plug in the given values into the equations:
For (1):
v1f = (44 kg * 80 m/s + 86 kg * (-24 m/s) - 86 kg * v2f) / 44 kg
For (2):
v2f = (44 kg * 80 m/s + 86 kg * (-24 m/s) - 44 kg * v1f) / 86 kg
Solving these equations will give us the final velocities:
To find the final velocities of the two balls after the collision, we can use the principles of conservation of momentum and kinetic energy.
(1) Final velocity of the 44 kg ball:
Let's assume the initial velocity of the 44 kg ball is v1 and the final velocity is v'1.
According to the principle of conservation of momentum, the total momentum before the collision should be equal to the total momentum after the collision.
Before the collision:
Momentum of the 44 kg ball = mass of 44 kg ball * initial velocity of 44 kg ball = 44 kg * 80 m/s = 3520 kg·m/s
Momentum of the 86 kg ball = mass of 86 kg ball * initial velocity of 86 kg ball = 86 kg * (-24 m/s) = -2064 kg·m/s (negative because it moves in the opposite direction)
Total momentum before the collision = 3520 kg·m/s + (-2064 kg·m/s) = 1456 kg·m/s
After the collision:
Momentum of the 44 kg ball = mass of 44 kg ball * final velocity of 44 kg ball = 44 kg * v'1
Momentum of the 86 kg ball = mass of 86 kg ball * final velocity of 86 kg ball = 86 kg * v'2 (assuming v'2 is the final velocity of the 86 kg ball)
Total momentum after the collision = (44 kg * v'1) + (86 kg * v'2)
Using the principle of conservation of momentum, we can set the total momentum before the collision equal to the total momentum after the collision:
1456 kg·m/s = (44 kg * v'1) + (86 kg * v'2) ---(equation 1)
Next, to determine the final velocities of the two balls, we can use the principle of conservation of kinetic energy. In an elastic collision, the kinetic energy is conserved.
Before the collision:
Kinetic energy of the 44 kg ball = (1/2) * mass of 44 kg ball * (initial velocity of 44 kg ball)^2 = (1/2) * 44 kg * (80 m/s)^2 = 140,800 J
Kinetic energy of the 86 kg ball = (1/2) * mass of 86 kg ball * (initial velocity of 86 kg ball)^2 = (1/2) * 86 kg * (-24 m/s)^2 = 24,768 J
Total kinetic energy before the collision = 140,800 J + 24,768 J = 165,568 J
After the collision:
Kinetic energy of the 44 kg ball = (1/2) * mass of 44 kg ball * (final velocity of 44 kg ball)^2 = (1/2) * 44 kg * v'1^2
Kinetic energy of the 86 kg ball = (1/2) * mass of 86 kg ball * (final velocity of 86 kg ball)^2 = (1/2) * 86 kg * v'2^2
Total kinetic energy after the collision = (1/2) * 44 kg * v'1^2 + (1/2) * 86 kg * v'2^2
Using the principle of conservation of kinetic energy, we can set the total kinetic energy before the collision equal to the total kinetic energy after the collision:
165,568 J = (1/2) * 44 kg * v'1^2 + (1/2) * 86 kg * v'2^2 ---(equation 2)
Now, we have two equations (equation 1 and equation 2) with two unknowns (v'1 and v'2). We can solve this system of equations to find the final velocities.