On a frictionless table, a 3kg block moving 4m/s to the right collides with an 8kg block moving 1.5m/s to the left. If the blocks stick together, what is the final velocity? How much mechanical energy is dissipate in the collision?
Where do I find the answers?????
To find the final velocity and the amount of mechanical energy dissipated in the collision, we can apply the principles of conservation of momentum and conservation of kinetic energy.
Step 1: Calculate the initial momentum of each block.
The initial momentum (p) of an object is given by the product of its mass (m) and velocity (v).
For the 3kg block moving to the right:
Initial momentum = mass x velocity
P1 = 3kg x 4m/s
P1 = 12 kg.m/s (to the right)
For the 8kg block moving to the left:
Initial momentum = mass x velocity (taking left as negative)
P2 = -8kg x (-1.5m/s)
P2 = 12 kg.m/s (to the right)
Step 2: Calculate the total initial momentum.
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
Total initial momentum = P1 + P2
Total initial momentum = 12 kg.m/s + 12 kg.m/s
Total initial momentum = 24 kg.m/s (to the right)
Step 3: Calculate the final velocity.
Since the blocks stick together after the collision, their final velocity is the same.
Total mass of the blocks after collision = 3kg + 8kg = 11kg
According to the principle of conservation of momentum, the total momentum after the collision is equal to the total momentum before the collision.
Total final momentum = Total initial momentum
Total mass after collision x final velocity = Total mass before collision x initial velocity
11kg x final velocity = 24 kg.m/s
final velocity = 24 kg.m/s / 11 kg
final velocity ≈ 2.18 m/s (to the right)
Therefore, the final velocity of the blocks after the collision is approximately 2.18 m/s to the right.
Step 4: Calculate the amount of mechanical energy dissipated.
The initial kinetic energy of the system is given by the sum of the kinetic energies of the individual blocks.
Kinetic energy (K) = (1/2)mv^2
Initial kinetic energy = (1/2)(3kg)(4m/s)^2 + (1/2)(8kg)(1.5m/s)^2
Initial kinetic energy = (1/2)(3kg)(16m^2/s^2) + (1/2)(8kg)(2.25m^2/s^2)
Initial kinetic energy = 24 J + 9 J
Initial kinetic energy = 33 J
The final kinetic energy of the system is given by the kinetic energy of the combined blocks after the collision.
Final kinetic energy = (1/2)(total mass)(final velocity)^2
Final kinetic energy = (1/2)(11kg)(2.18m/s)^2
Final kinetic energy = (1/2)(11kg)(4.7524m^2/s^2)
Final kinetic energy ≈ 26.08 J
The amount of mechanical energy dissipated in the collision is equal to the difference between the initial kinetic energy and the final kinetic energy.
Amount of energy dissipated = Initial kinetic energy - Final kinetic energy
Amount of energy dissipated = 33 J - 26.08 J
Amount of energy dissipated ≈ 6.92 J
Therefore, approximately 6.92 J of mechanical energy is dissipated in the collision.
To find the final velocity and the amount of mechanical energy dissipated in the collision, we first need to apply the principle of conservation of momentum and conservation of mechanical energy.
1. Conservation of Momentum:
The principle of conservation of momentum states that the total momentum of a system remains constant before and after a collision, provided that no external forces act on the system. Mathematically, it can be expressed as:
Initial momentum = Final momentum
Momentum (p) = mass (m) * velocity (v)
Let's assign a positive direction to the right and a negative direction to the left.
The initial momentum is calculated by summing the individual momenta of the two blocks before the collision:
Initial momentum = (mass of the 3kg block * velocity of the 3kg block) + (mass of the 8kg block * velocity of the 8kg block)
Initial momentum = (3 kg * 4 m/s) + (8 kg * (-1.5 m/s))
2. Conservation of Mechanical Energy:
Since the collision is said to be completely inelastic, where the two blocks stick together, we cannot use the principle of conservation of kinetic energy. However, we can calculate the initial and final kinetic energy to find the mechanical energy dissipated in the collision.
Initial kinetic energy = (1/2) * mass * velocity^2
Final kinetic energy = (1/2) * (combined mass) * (final velocity)^2
Now, let's calculate the results.
Step 1: Calculate the initial momentum:
Initial momentum = (3 kg * 4 m/s) + (8 kg * (-1.5 m/s))
Step 2: Calculate the combined mass and momentum after the collision:
Combined mass = mass of the 3kg block + mass of the 8kg block
Final momentum = combined mass * final velocity
Step 3: Calculate the final velocity by rearranging the equation for final momentum:
Final velocity = Final momentum / combined mass
Step 4: Calculate the initial and final kinetic energy:
Initial kinetic energy = (1/2) * (mass of the 3kg block * velocity of the 3kg block^2) + (1/2) * (mass of the 8kg block * velocity of the 8kg block^2)
Final kinetic energy = (1/2) * (combined mass * final velocity^2)
Step 5: Calculate the mechanical energy dissipated:
Mechanical energy dissipated = Initial kinetic energy - Final kinetic energy
By following these steps, you should be able to find the final velocity and the amount of mechanical energy dissipated in the collision.
+x to the right
initial momentum = 3*4 - 8*1.5 = 0
so final momentum = 0
so final velocity = 0
energy is all gone
Initial Ke = (1/2)(3)(16) + (1/2)(8)(2.25)