A 5.25-kg ball, moving to the right at a velocity of +2.10 m/s on a frictionless table, collides head-on with a stationary 7.60-kg ball. Find the final velocities of the balls if the collision meet the following conditions.
(a) elastic
5.25-kg ball = Incorrect: Your answer is incorrect. m/s
7.6-kg ball = m/s
(b) completely inelastic
momentum
5.25*2.1+O=(5.25)V1+7.60(V2)
energy:
1/2 5.25*2.1+0=1/2 5.25v1^2 + 1/2 7.6v2^2
or 5.25*2.1+0= 5.25v1^2 + 7.6v2^2
In the first equation:
v2=(5.25(2.1-V1)/7.6
so now, square that
v2^2=.69^2(2.1^2-4.2v1+vi^2)
and put it into
5.25*2.1+0=1/2 5.25v1^2 + 7.6v2^2
or
5.25*2.1+0=1/2 5.25v1^2 + 7.6.69^2(2.1^2-4.2v1+vi^2)
Now, do the algebra and get it to a quadratic form and find the two solutions for v1.
Then go back and find V2
To find the final velocities of the balls after the collision, we can use the principles of conservation of momentum and kinetic energy.
(a) In an elastic collision, both momentum and kinetic energy are conserved.
First, we calculate the initial momentum of the system:
Initial momentum of ball 1 = mass of ball 1 * velocity of ball 1
= 5.25 kg * 2.10 m/s
= 11.025 kg·m/s
Initial momentum of ball 2 = mass of ball 2 * velocity of ball 2
= 7.60 kg * 0 m/s (as it is stationary)
= 0 kg·m/s
Total initial momentum = Initial momentum of ball 1 + Initial momentum of ball 2
= 11.025 kg·m/s + 0 kg·m/s
= 11.025 kg·m/s
Since momentum is conserved, the total final momentum will also be 11.025 kg·m/s.
Next, we calculate the initial kinetic energy of the system:
Initial kinetic energy of ball 1 = (1/2) * mass of ball 1 * (velocity of ball 1)^2
= (1/2) * 5.25 kg * (2.10 m/s)^2
= 11.00225 J
Initial kinetic energy of ball 2 = (1/2) * mass of ball 2 * (velocity of ball 2)^2
= (1/2) * 7.60 kg * (0 m/s)^2
= 0 J
Total initial kinetic energy = Initial kinetic energy of ball 1 + Initial kinetic energy of ball 2
= 11.00225 J + 0 J
= 11.00225 J
Since kinetic energy is conserved in an elastic collision, the total final kinetic energy will also be 11.00225 J.
Now, let's assume the final velocities of the balls after the collision are v1f and v2f.
Final momentum of ball 1 = mass of ball 1 * final velocity of ball 1
= 5.25 kg * v1f
Final momentum of ball 2 = mass of ball 2 * final velocity of ball 2
= 7.60 kg * v2f
Total final momentum = Final momentum of ball 1 + Final momentum of ball 2
= 5.25 kg * v1f + 7.60 kg * v2f
Since momentum conservation tells us that the total final momentum is equal to the total initial momentum (11.025 kg·m/s), we can write the equation:
5.25 kg * v1f + 7.60 kg * v2f = 11.025 kg·m/s
Similarly, the final kinetic energy of the system can be written as:
Final kinetic energy = (1/2) * mass of ball 1 * (final velocity of ball 1)^2 + (1/2) * mass of ball 2 * (final velocity of ball 2)^2
Since kinetic energy conservation tells us that the total final kinetic energy is equal to the total initial kinetic energy (11.00225 J), we can write the equation:
(1/2) * 5.25 kg * (v1f)^2 + (1/2) * 7.60 kg * (v2f)^2 = 11.00225 J
Now, we have a system of two equations with two unknowns (v1f and v2f). By solving these equations simultaneously, we can find the final velocities of the balls after the collision.
(b) In a completely inelastic collision, the two balls stick together after the collision and move as a single unit with a common final velocity.
To find the final velocity in a completely inelastic collision, we can use the principle of conservation of momentum.
Initial momentum of the system = Total mass of both balls * common final velocity
Total mass of both balls = mass of ball 1 + mass of ball 2
= 5.25 kg + 7.60 kg
= 12.85 kg
Initial momentum of the system = 12.85 kg * common final velocity
Since momentum is conserved, the initial momentum of the system will be equal to the final momentum of the system:
12.85 kg * common final velocity = Initial momentum of the system
= 11.025 kg·m/s (using the value we calculated in part (a))
From this equation, we can solve for the common final velocity.