A 4.63-g bullet is moving horizontally with a velocity of +349 m/s, where the sign + indicates that it is moving to the right (see part a of the drawing). The bullet is approaching two blocks resting on a horizontal frictionless surface. Air resistance is negligible. The bullet passes completely through the first block (an inelastic collision) and embeds itself in the second one, as indicated in part b. Note that both blocks are moving after the collision with the bullet. The mass of the first block is 1223 g, and its velocity is +0.725 m/s after the bullet passes through it. The mass of the second block is 1628 g. (a) What is the velocity of the second block after the bullet imbeds itself? (b) Find the ratio of the total kinetic energy after the collision to that before the collision.
To find the velocity of the second block after the bullet imbeds itself, we will use the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum of an isolated system remains constant if no external forces act on it.
We can calculate the velocity of the second block after the collision using the formula:
(m1 * v1 + m2 * v2) = (m1 * u1 + m2 * u2),
where:
- m1 and m2 are the masses of the first and second blocks, respectively,
- v1 and v2 are the velocities of the first and second blocks, respectively, after the collision,
- u1 is the velocity of the first block before the collision (0.725 m/s in this case), and
- u2 is the velocity of the bullet before the collision (349 m/s in this case).
Substituting the given values into the equation:
(1223 g * 0.725 m/s) + (1628 g * v2) = (1223 g * 0.725 m/s) + (1628 g * 349 m/s),
Converting the masses to kilograms and solving for v2:
(1.223 kg * 0.725 m/s) + (1.628 kg * v2) = (1.223 kg * 0.725 m/s) + (1.628 kg * 349 m/s),
0.888275 kg·m/s + (1.628 kg * v2) = 0.888275 kg·m/s + (568.172 kg·m/s),
1.628 kg * v2 = 568.172 kg·m/s,
v2 = 568.172 kg·m/s / 1.628 kg,
v2 ≈ 349.24 m/s.
Therefore, the velocity of the second block after the bullet imbeds itself is approximately 349.24 m/s.
Now, let's move on to finding the ratio of the total kinetic energy after the collision to that before the collision.
The total kinetic energy after the collision can be calculated as the sum of the kinetic energies of both blocks. The formula for kinetic energy is given by:
KE = (1/2) * m * v^2,
where m is the mass of the object and v is its velocity.
The total kinetic energy before the collision can be calculated using the same formula.
Using this information, we have:
KE_total_after = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2,
KE_total_before = (1/2) * m1 * u1^2 + (1/2) * m2 * u2^2.
Substituting the given values into the equations:
KE_total_after = (1/2) * 1223 g * (0.725 m/s)^2 + (1/2) * 1628 g * (349.24 m/s)^2,
KE_total_before = (1/2) * 1223 g * (0.725 m/s)^2 + (1/2) * 1628 g * (349 m/s)^2.
Calculating the kinetic energy values:
KE_total_after = 188.67 J + 101836.27 J,
KE_total_before = 188.67 J + 101657.62 J.
Therefore, the ratio of the total kinetic energy after the collision to that before the collision can be calculated as follows:
KE_ratio = KE_total_after / KE_total_before,
KE_ratio = (188.67 J + 101836.27 J) / (188.67 J + 101657.62 J),
KE_ratio ≈ 1.0018.
Hence, the ratio of the total kinetic energy after the collision to that before the collision is approximately 1.0018.
To solve this problem, we can apply the principle of conservation of momentum and conservation of kinetic energy.
(a) To find the velocity of the second block after the bullet embeds itself, we need to use the principle of conservation of momentum. The total momentum before the collision should be equal to the total momentum after the collision.
Before the collision:
The bullet has a mass of 4.63 g (0.00463 kg) and a velocity of +349 m/s. Therefore, its momentum is:
Momentum_bullet_before = mass_bullet * velocity_bullet = 0.00463 kg * 349 m/s
The first block has a mass of 1223 g (1.223 kg) and a velocity of +0.725 m/s. Therefore, its momentum is:
Momentum_block1_before = mass_block1 * velocity_block1 = 1.223 kg * 0.725 m/s
The total momentum before the collision is the sum of the individual momenta:
Total_momentum_before = Momentum_bullet_before + Momentum_block1_before
After the collision:
After the bullet embeds in the second block, the two blocks move together with a common velocity. Let's assume this final velocity as v2.
The second block has a mass of 1628 g (1.628 kg). Therefore, its momentum after the collision is:
Momentum_block2_after = mass_block2 * velocity_block2 = 1.628 kg * v2
The total momentum after the collision should be equal to the total momentum before the collision:
Total_momentum_before = Total_momentum_after
Setting up the equation:
Total_momentum_before = Total_momentum_after
Momentum_bullet_before + Momentum_block1_before = Momentum_block2_after
Substituting the known values, we have:
0.00463 kg * 349 m/s + 1.223 kg * 0.725 m/s = 1.628 kg * v2
Simplifying the equation and solving for v2:
v2 = (0.00463 kg * 349 m/s + 1.223 kg * 0.725 m/s) / 1.628 kg
Calculate the value of v2 using the given masses and velocities.
(b) To find the ratio of the total kinetic energy after the collision to that before the collision, we need to calculate the total kinetic energy in both situations.
Before the collision:
The total kinetic energy before the collision is the sum of the individual kinetic energies. The kinetic energy of an object is given by the equation: KE = 0.5 * mass * velocity^2.
The kinetic energy of the bullet before the collision is:
KE_bullet_before = 0.5 * mass_bullet * velocity_bullet^2
The kinetic energy of the first block before the collision is:
KE_block1_before = 0.5 * mass_block1 * velocity_block1^2
Total_KE_before = KE_bullet_before + KE_block1_before
After the collision:
The total kinetic energy after the collision is the sum of the kinetic energy of the two blocks moving together.
The kinetic energy of the second block after the collision is:
KE_block2_after = 0.5 * mass_block2 * velocity_block2^2
Total_KE_after = KE_block2_after
The ratio of the total kinetic energy after the collision to that before the collision is:
Ratio_KE = Total_KE_after / Total_KE_before
Calculate the values of Total_KE_before and Total_KE_after using the given masses and velocities. Then calculate the ratio of the total kinetic energy.