A puck of mass 0.60 kg approaches a second, identical puck that is stationary on frictionless ice. The initial speed of the moving puck is 5.0 m/s. After the collision, one puck leaves with a speed v1 at 30° to the original line of motion. The second puck leaves with speed v2 at 60°.
Calculate v1 and v2.
You have two equations of momentum conservation to work with: One for the original direction of motion and another for the second, perpendicular, direction.
The two pucks must leave in opposite sides of the original line of motion.
The M's are the same and cancel out of the momentum equations. Thus
v1 sin 30 - v2 sin 60 = 0
v1 cos 30 + v2 cos 60 = Vo = 5.0 m/s
v1 - v2*sqrt3 = 0
v1*sqrt3 + v2 = 10
3*v1+ v2*sqrt3 = 10 sqrt3
4*v1 = 10*sqrt3
v1 = (5/2)*sqrt3 = 4.33 m/s
v2 = v1/sqrt3 = 2.50 m/s
Kinetic energy happens to be conserved in this case, but I did not have to use that fact to solve the problem. You will note that
(M/2)v1^2 + (M/2)v2^2 = (M/2)Vo^2.
To solve this problem, we can use the principle of conservation of momentum. According to the principle of conservation of momentum, the total momentum before the collision should be equal to the total momentum after the collision.
Step 1: Calculate the initial momentum of the system before the collision.
The initial momentum (p_initial) can be calculated by multiplying the mass (m) of the puck by its initial velocity (v_initial).
p_initial = m * v_initial
Given that the mass (m) of the puck is 0.60 kg and the initial velocity (v_initial) is 5.0 m/s:
p_initial = 0.60 kg * 5.0 m/s
Step 2: Calculate the final momentum of the system after the collision.
The final momentum (p_final) can be calculated by adding the individual momenta of the two pucks.
p_final = m1 * v1 + m2 * v2
Given that the masses (m1 and m2) of both pucks are equal (0.60 kg):
p_final = 0.60 kg * v1 + 0.60 kg * v2
Step 3: Apply the principle of conservation of momentum.
According to the conservation of momentum principle, the initial momentum should be equal to the final momentum.
p_initial = p_final
Substituting the values from Step 1 and Step 2 into this equation, we get:
0.60 kg * 5.0 m/s = 0.60 kg * v1 + 0.60 kg * v2
Step 4: Solve for v1 and v2.
To solve for v1 and v2, we need another equation. We can use the conservation of energy principle, which states that the kinetic energy before the collision should be equal to the kinetic energy after the collision.
Step 4.1: Calculate the initial kinetic energy of the system before the collision.
The initial kinetic energy (KE_initial) can be calculated using the mass (m) and initial velocity (v_initial) of the puck.
KE_initial = (1/2) * m * v_initial^2
Given that the mass (m) of the puck is 0.60 kg and the initial velocity (v_initial) is 5.0 m/s:
KE_initial = (1/2) * 0.60 kg * (5.0 m/s)^2
Step 4.2: Calculate the final kinetic energy of the system after the collision.
The final kinetic energy (KE_final) can be calculated using the masses (m1 and m2) and velocities (v1 and v2) of the respective pucks.
KE_final = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2
Given that the masses (m1 and m2) of both pucks are equal (0.60 kg), and the angles (30° and 60°) are given:
KE_final = (1/2) * 0.60 kg * v1^2 + (1/2) * 0.60 kg * v2^2
Step 4.3: Apply the conservation of energy principle.
According to the conservation of energy principle, the initial kinetic energy should be equal to the final kinetic energy.
KE_initial = KE_final
Substituting the values from Step 4.1 and Step 4.2 into this equation, we get:
(1/2) * 0.60 kg * (5.0 m/s)^2 = (1/2) * 0.60 kg * v1^2 + (1/2) * 0.60 kg * v2^2
Step 5: Solve for v1 and v2.
Now, we have two equations:
1) 0.60 kg * 5.0 m/s = 0.60 kg * v1 + 0.60 kg * v2
2) (1/2) * 0.60 kg * (5.0 m/s)^2 = (1/2) * 0.60 kg * v1^2 + (1/2) * 0.60 kg * v2^2
We can solve these two equations simultaneously to find the values of v1 and v2.
To calculate the final speeds (v1 and v2) of the two pucks after the collision, we can use conservation of momentum and conservation of kinetic energy.
1. Conservation of Momentum:
The total momentum before the collision is equal to the total momentum after the collision. The momentum (p) of an object is given by its mass (m) multiplied by its velocity (v).
So, we have:
Initial momentum = Final momentum
The initial momentum of the system is:
p_initial = m1 * v1_initial + m2 * v2_initial
where m1 and m2 are the masses of the two pucks, v1_initial is the initial velocity of the first puck, and v2_initial is the initial velocity of the second puck.
Since the second puck is initially stationary (v2_initial = 0), the equation simplifies to:
p_initial = m1 * v1_initial
The final momentum of the system is:
p_final = m1 * v1_final + m2 * v2_final
where v1_final and v2_final are the final velocities of the first and second pucks, respectively.
Using the conservation of momentum, we can equate the initial momentum to the final momentum:
m1 * v1_initial = m1 * v1_final + m2 * v2_final
2. Conservation of Kinetic Energy:
The total kinetic energy before the collision is equal to the total kinetic energy after the collision. The kinetic energy (K) of an object is given by one-half times its mass times the square of its velocity.
So, we have:
Initial kinetic energy = Final kinetic energy
The initial kinetic energy of the system is:
K_initial = (1/2) * m1 * v1_initial^2 + (1/2) * m2 * v2_initial^2
The final kinetic energy of the system is:
K_final = (1/2) * m1 * v1_final^2 + (1/2) * m2 * v2_final^2
Using the conservation of kinetic energy, we can equate the initial kinetic energy to the final kinetic energy:
(1/2) * m1 * v1_initial^2 = (1/2) * m1 * v1_final^2 + (1/2) * m2 * v2_final^2
Now we have two equations with two unknowns: v1_final and v2_final.
Solving these equations simultaneously will give us the values of v1_final and v2_final.
Plug in the values:
m1 = 0.60 kg (mass of each puck)
v1_initial = 5.0 m/s
From the given information, m2 and the angles at which the pucks leave are not relevant to finding v1 and v2.
Simplify the equations:
0.60 * 5.0 = 0.60 * v1_final + 0.60 * v2_final ----(1)
(1/2) * 0.60 * 5.0^2 = (1/2) * 0.60 * v1_final^2 + (1/2) * 0.60 * v2_final^2 ----(2)
Solve the equations using simultaneous equations or substitution to find the values of v1_final and v2_final.