Two coupled boxcars are rolling along at 4.0 when they collide with and couple to a third, stationary boxcar.
1.) What is the final speed of the three coupled boxcars?
2.) What fraction of the cars' initial kinetic energy is transformed into thermal energy?
To solve these questions, we can use the principle of conservation of momentum and kinetic energy.
1.) What is the final speed of the three coupled boxcars?
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is given by the product of its mass and velocity, so we can write:
(m1 * v1) + (m2 * v2) = (m1 + m2 + m3) * vf
Where m1, m2, and m3 are the masses of the first, second, and third boxcars respectively, v1 and v2 are their initial velocities, and vf is the final velocity.
Since the first and second boxcars have the same velocity (4.0 m/s) and are coupled together, we can write:
(2m * 4.0) + (m * 0) = (2m + m) * vf
Simplifying the equation:
(8m) = (3m) * vf
Dividing both sides by 3m:
vf = 8/3 = 2.67 m/s
Therefore, the final speed of the three coupled boxcars is 2.67 m/s.
2.) What fraction of the cars' initial kinetic energy is transformed into thermal energy?
The initial kinetic energy of the system can be calculated using the equation:
KE_initial = (0.5 * m1 * v1^2) + (0.5 * m2 * v2^2)
The final kinetic energy of the system can be calculated using the equation:
KE_final = 0.5 * (m1 + m2 + m3) * vf^2
The fraction of the initial kinetic energy transformed into thermal energy can be calculated using the equation:
Fraction = (KE_initial - KE_final) / KE_initial
Substituting the given values:
KE_initial = (0.5 * 2m * 4.0^2) + (0.5 * m * 4.0^2)
= 16m^2 + 8m^2
= 24m^2
KE_final = 0.5 * (2m + m + m) * 2.67^2
= 2.67^2 * 4m
= 28.4556m
Fraction = (24m^2 - 28.4556m) / 24m^2
Simplifying the equation:
Fraction = (24 - 28.4556) / 24
= -4.4556/24
= -0.18565
The negative value indicates that energy was lost as thermal energy. Therefore, approximately 18.57% of the cars' initial kinetic energy is transformed into thermal energy.
To answer these questions, we can use the principle of conservation of momentum and the principle of conservation of energy.
1.) Final Speed of the Three Coupled Boxcars:
According to the principle of conservation of momentum, the total momentum before the collision must be equal to the total momentum after the collision.
Momentum (p) is defined as the product of mass (m) and velocity (v): p = m * v.
The total momentum before the collision is the sum of the individual momenta of the two boxcars initially in motion. Let's denote the mass and velocity of the first boxcar as m1 and v1, and that of the second boxcar as m2 and v2.
Before the collision:
Total momentum = momentum of first boxcar + momentum of second boxcar
= (m1 * v1) + (m2 * v2)
After the collision, the three boxcars are coupled together and have a common final velocity, which we denote as vf. Since they are coupled, we can treat them as a single system. The total momentum after the collision is given by the product of the total mass of the three boxcars, denoted as M, and their final velocity vf.
After the collision:
Total momentum = M * vf
Using the principle of conservation of momentum, we can equate the total momentum before the collision to the total momentum after the collision:
(m1 * v1) + (m2 * v2) = M * vf
We know the values of m1, v1, m2, and v2, so we can substitute these values and solve for vf to get the final speed of the three coupled boxcars.
2.) Fraction of Initial Kinetic Energy Transformed into Thermal Energy:
To determine this, we need to calculate the initial total kinetic energy and the final total thermal energy.
The total kinetic energy before the collision is the sum of the individual kinetic energies of the two boxcars initially in motion. The kinetic energy (K) is given by the formula: K = (1/2) * m * v^2.
Before the collision:
Total initial kinetic energy = kinetic energy of first boxcar + kinetic energy of second boxcar
= (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2
After the collision, some portion of the initial kinetic energy is transformed into thermal energy due to deformation and internal friction within the boxcars. We denote the final total thermal energy as T.
After the collision:
Total final thermal energy = T
The fraction of the initial kinetic energy transformed into thermal energy is given by:
Fraction = (Total initial kinetic energy - Total final thermal energy) / Total initial kinetic energy
Substituting the values of the initial kinetic energies and final thermal energy, we can calculate this fraction.
Please provide the masses and velocities of the boxcars involved in order to compute the final speed and fraction of energy transformed into thermal energy.
Assume that the boxcars all have the same mass.
Use conservation of momentum:
2*m*vi = 3*m*vf
where m is the mass of one boxcar, vi is the initial speed, vf is the final speed.
2*m*4 = 3*m*vf
vf = 8/3
Initial kinetic energy is ki
1/2*(2*m)*vi^2 = m*4^2 = 16*m
Final kinetic energy kf is
1/2*(3*m)*(8/3)^2 = 1/2*(3*m)*(64/9) = (32/3)*m
The fraction of the car's initial kinetic energy that is transformed into thermal energy is (ki - kf)/ki