A 2.0 kg mass traveling at 3.0 m/s strikes another 2.0 kg mass traveling at -1.0 m/s. They collide and have a complete elastic collision. If the second mass moves at 3.0 m/s after the collision, what does the first mass do?
momentum is conserved
so they "trade" velocities
To determine what the first mass does after the collision, we need to apply the principles of conservation of momentum and kinetic energy.
Conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision.
The equation for momentum is:
momentum (p) = mass (m) × velocity (v)
For the first mass:
Initial momentum = 2.0 kg × 3.0 m/s = 6.0 kg·m/s
After the collision, the total momentum is the sum of the individual momenta of the two masses:
Total momentum = momentum of first mass + momentum of second mass
Let's call the velocity of the first mass after the collision v1'.
Since the collision is elastic, the second mass also has a momentum of 6.0 kg·m/s. The equation for the second mass is:
6.0 kg·m/s = 2.0 kg × 3.0 m/s + 2.0 kg × v1'
Now we can solve for v1':
6.0 kg·m/s - 2.0 kg × 3.0 m/s = 2.0 kg × v1'
6.0 kg·m/s - 6.0 kg·m/s = 2.0 kg × v1'
0 = 2.0 kg × v1'
Since the second mass has the same velocity (3.0 m/s) after the collision, the first mass will come to a stop after the collision and remain stationary.
To determine the motion of the first mass after the collision, we can use the principle of conservation of linear momentum. In an elastic collision, both the total momentum and the total kinetic energy of the system are conserved.
In this case, we have two masses colliding with each other. The momentum of an object is the product of its mass and velocity. The initial momentum of the system is given by:
P_initial = m1 * v1_initial + m2 * v2_initial
Where:
m1 = mass of the first mass = 2.0 kg
v1_initial = initial velocity of the first mass = 3.0 m/s
m2 = mass of the second mass = 2.0 kg
v2_initial = initial velocity of the second mass = -1.0 m/s
Substituting the given values, we can calculate the initial momentum of the system:
P_initial = (2.0 kg)(3.0 m/s) + (2.0 kg)(-1.0 m/s)
= 6.0 kg·m/s - 2.0 kg·m/s
= 4.0 kg·m/s
Now, let's assume that after the collision, the first mass moves with a velocity v1_final, while the second mass moves with a velocity v2_final of 3.0 m/s.
The final momentum of the system can then be calculated as:
P_final = m1 * v1_final + m2 * v2_final
Using the conservation of linear momentum, we can say that P_initial = P_final. Therefore:
4.0 kg·m/s = (2.0 kg) * v1_final + (2.0 kg) * (3.0 m/s)
Simplifying the equation:
4.0 kg·m/s = 2.0 kg·v1_final + 6.0 kg·m/s
Rearranging the equation to solve for v1_final:
(2.0 kg) * v1_final = 4.0 kg·m/s - 6.0 kg·m/s
v1_final = -1.0 kg·m/s
Therefore, the first mass will move in the opposite direction with a velocity of 1.0 m/s.