A ball of mass m1= .250 kg and velocity v01 = + 5 m/s collides head-on with a ball of mass m2= .800 kg that is initially at rest (v02 = 0 m/s). No external forces act on the balls. If the collision is elastic, what are the velocities of the balls after the collision?
The formulas that you need to use for the final velocities can be found here:
http://hyperphysics.phy-astr.gsu.edu/hbase/elacol2.html
The head-on collision case is in part "c"
To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.
Conservation of momentum states that the total momentum of a system before a collision is equal to the total momentum after the collision. Mathematically, it can be expressed as:
m1*v01 + m2*v02 = m1*v1 + m2*v2
where m1 and m2 are the masses of the balls, v01 and v02 are their initial velocities, and v1 and v2 are their final velocities.
In this case, since ball 2 is initially at rest (v02 = 0 m/s), the equation simplifies to:
m1*v01 = m1*v1 + m2*v2
Next, we can use the conservation of kinetic energy, which states that the total kinetic energy before the collision is equal to the total kinetic energy after the collision. Mathematically, it can be expressed as:
(1/2)*m1*(v01)^2 + (1/2)*m2*(v02)^2 = (1/2)*m1*(v1)^2 + (1/2)*m2*(v2)^2
Substituting v02 = 0, this equation simplifies to:
(1/2)*m1*(v01)^2 = (1/2)*m1*(v1)^2 + (1/2)*m2*(v2)^2
Now we have two equations with two unknowns (v1 and v2). We can solve these equations simultaneously to find the final velocities.
Substituting v02 = 0 and v01 = 5 m/s into the first equation:
0.250 kg * 5 m/s = 0.250 kg * v1 + 0.800 kg * v2
1.25 kg m/s = 0.250 kg * v1 + 0.800 kg * v2 (equation 1)
Substituting v02 = 0 and v01 = 5 m/s into the second equation:
(1/2) * 0.250 kg * (5 m/s)^2 = (1/2) * 0.250 kg * (v1)^2 + (1/2) * 0.800 kg * (v2)^2
6.25 J = 0.125 kg * (v1)^2 + 0.400 kg * (v2)^2 (equation 2)
Now we have a system of linear equations that can be solved algebraically. By rearranging equation 1, we can express it as:
v1 = (1.25 kg m/s - 0.800 kg * v2) / 0.250 kg
Substituting this expression for v1 into equation 2, we can solve for v2:
6.25 J = 0.125 kg * [(1.25 kg m/s - 0.800 kg * v2) / 0.250 kg]^2 + 0.400 kg * (v2)^2
Simplifying this equation will give us the value of v2. Then, substituting the value of v2 into equation 1 will give us the value of v1.