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.
Oh, bullets and blocks, sounds like quite the action movie! Let's dive into your questions, shall we?
(a) To find the velocity of the second block after the bullet embeds itself, we need to use the principle of conservation of momentum. Since there is no external force acting on the system, the total momentum before the collision should equal the total momentum after the collision.
The momentum of the bullet before the collision is given by:
Momentum = mass × velocity
= 4.63 g × 349 m/s
Now, after the inelastic collision, the bullet embeds itself in the second block, so the momentum of the second block with the bullet inside is given by:
Momentum = (mass of the second block + mass of the bullet) × velocity of the second block
We can now set up an equation:
momentum before collision = momentum after collision
(4.63 g × 349 m/s) = (1628 g + 4.63 g) × velocity of the second block
Now we can solve for the velocity of the second block.
(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 before and after the collision.
Before the collision, the total kinetic energy is the sum of the kinetic energy of the bullet and the first block:
Total kinetic energy before collision = (1/2) × mass of the bullet × (velocity of the bullet)^2 + (1/2) × mass of the first block × (velocity of the first block)^2
After the collision, the bullet is embedded in the second block, so the total kinetic energy is the sum of the kinetic energy of the second block and the bullet inside:
Total kinetic energy after collision = (1/2) × (mass of the second block + mass of the bullet) × (velocity of the second block)^2
To find the ratio, simply divide the total kinetic energy after the collision by the total kinetic energy before the collision.
I hope this helps! Don't hesitate to reach out if you have more physics questions or need a good joke to lighten the mood!
To solve this problem, we will 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 can use the conservation of momentum.
The initial momentum before the collision is given by:
Initial momentum = (mass of bullet) * (velocity of bullet) + (mass of first block) * (velocity of first block)
Initial momentum = (4.63 g) * (349 m/s) + (1223 g) * (0 m/s) = 1613.87 g·m/s
(Note: 1 kg = 1000 g)
Since the bullet embeds itself in the second block, the total mass after the collision is the sum of the masses of the bullet and the second block.
Total mass after the collision = (mass of bullet) + (mass of second block)
Total mass after the collision = 4.63 g + 1628 g = 1632.63 g
Now, let's find the velocity of the second block after the collision, which we'll denote as v_2.
Conservation of momentum equation:
(initial momentum) = (final momentum)
(1613.87 g·m/s) = (1632.63 g) * (v_2)
Solving for v_2:
v_2 = (1613.87 g·m/s) / (1632.63 g)
v_2 ≈ 0.988 m/s
Therefore, the velocity of the second block after the bullet embeds itself is approximately +0.988 m/s to the right.
(b) To find the ratio of the total kinetic energy after the collision to that before the collision, we need to calculate both.
The initial kinetic energy before the collision is given by:
Initial kinetic energy = 0.5 * (mass of bullet) * (velocity of bullet)^2 + 0.5 * (mass of first block) * (velocity of first block)^2
Initial kinetic energy = 0.5 * (4.63 g) * (349 m/s)^2 + 0.5 * (1223 g) * (0.725 m/s)^2
Initial kinetic energy ≈ 279,668.4 g·m^2/s^2
The final kinetic energy after the collision is given by:
Final kinetic energy = 0.5 * (total mass after the collision) * (velocity of second block after collision)^2
Final kinetic energy = 0.5 * (1632.63 g) * (0.988 m/s)^2
Final kinetic energy ≈ 808.32 g·m^2/s^2
Now, we can calculate the ratio of the total kinetic energy after the collision to that before the collision:
Ratio = (Final kinetic energy) / (Initial kinetic energy)
Ratio = (808.32 g·m^2/s^2) / (279,668.4 g·m^2/s^2)
Ratio ≈ 0.002895
Therefore, the ratio of the total kinetic energy after the collision to that before the collision is approximately 0.002895.
To answer the questions, we need to apply the principle of conservation of momentum and the principle of conservation of kinetic energy.
(a) To find the velocity of the second block after the bullet embeds itself, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
The initial momentum is given by the product of the mass and velocity of the bullet:
Initial momentum = mass of bullet * velocity of bullet
The final momentum is the sum of the momentum of the first block and the second block:
Final momentum = (mass of first block * velocity of first block) + (mass of second block * velocity of second block)
Since the bullet embeds itself in the second block, their momenta are combined.
Setting the initial momentum equal to the final momentum:
Initial momentum = Final momentum
mass of bullet * velocity of bullet = (mass of first block * velocity of first block) + (mass of second block * velocity of second block)
Plugging in the given values:
(4.63 g) * (+349 m/s) = (1223 g) * (+0.725 m/s) + (1628 g) * velocity of second block
Solving for the velocity of the second block:
velocity of second block = [(4.63 g) * (+349 m/s) - (1223 g) * (+0.725 m/s)] / (1628 g)
Calculating the velocity of the second block will give us the answer to part (a).
(b) To find the ratio of the total kinetic energy after the collision to that before the collision, we need to calculate the initial and final kinetic energies.
The initial kinetic energy is given by:
Initial kinetic energy = (1/2) * mass of bullet * (velocity of bullet)^2
The final kinetic energy is the sum of the kinetic energies of the first and second blocks:
Final kinetic energy = (1/2) * mass of first block * (velocity of first block)^2 + (1/2) * mass of second block * (velocity of second block)^2
Dividing the final kinetic energy by the initial kinetic energy will give us the ratio of the total kinetic energy after the collision to that before the collision.
Calculating the ratio will provide the answer to part (b).