A(n) 19 g object moving to the right at
39 cm/s overtakes and collides elastically with
a 36 g object moving in the same direction at
19 cm/s.
Find the velocity of the slower object after
the collision.
Answer in units of cm/s.
To find the velocity of the slower object after the collision, we can use the principles of conservation of momentum and conservation of kinetic energy.
Conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. Mathematically, we can express this as:
m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final
Where:
- m1 and m2 are the masses of the objects
- v1_initial and v2_initial are the initial velocities of the objects
- v1_final and v2_final are the final velocities of the objects
In this case, object 1 has mass m1 = 19 g and object 2 has mass m2 = 36 g. The initial velocities are v1_initial = 39 cm/s for object 1 and v2_initial = 19 cm/s for object 2.
Now, let's solve for the final velocity v2_final of the slower object (object 2):
m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final
(19 g * 39 cm/s) + (36 g * 19 cm/s) = (19 g * v1_final) + (36 g * v2_final)
Now we can substitute the given values:
(19 * 39) + (36 * 19) = (19 * v1_final) + (36 * v2_final)
741 + 684 = 19v1_final + 36v2_final
Simplifying the equation:
1425 = 19v1_final + 36v2_final
We also know that the collision is elastic, which means that kinetic energy is conserved. Mathematically, this can be expressed as:
(1/2) * m1 * v1_initial^2 + (1/2) * m2 * v2_initial^2 = (1/2) * m1 * v1_final^2 + (1/2) * m2 * v2_final^2
Plugging in the known values:
(1/2) * 19 g * (39 cm/s)^2 + (1/2) * 36 g * (19 cm/s)^2 = (1/2) * 19 g * v1_final^2 + (1/2) * 36 g * v2_final^2
Simplifying the equation:
(1/2) * 19 * (39)^2 + (1/2) * 36 * (19)^2 = (1/2) * 19 * v1_final^2 + (1/2) * 36 * v2_final^2
Now we have a system of equations that can be solved simultaneously:
1425 = 19v1_final + 36v2_final
(1/2) * (19) * (39^2) + (1/2) * (36) * (19^2) = (1/2) * (19) * (v1_final^2) + (1/2) * (36) * (v2_final^2)
Let's solve the system of equations to find the final velocity v2_final.
in these odd units
momentum before = 19(39) + 36 (19) = 741+684 = 1425
momentum after = 19 Va + 36 Vb = 1425
Ke before = .5 * 19*39^2 + .5*36*19^2 = Ke after = .5*19 Va^2 + .5*36 Vb^2