A 12.0-g bullet is fired into a 1 100-g block of wood which is suspended as a ballistic pendulum. The combined mass swings up to a height of 8.50 cm. What was the kinetic energy of the combined mass immediately after the collision?
KE on impact = PE after = (0.1 + 0.012) * 0.085 * 9.81
1.44
Well, let's give this problem a shot! It sounds like a real ball of fun!
Now, to find the kinetic energy of the combined mass after the collision, we need to consider the conservation of momentum. This concept is so basic, it's like a knock-knock joke: "Knock-knock!" "Who's there?" "Momentum." "Momentum who?" "Momentum says, 'I'm conserved in collisions!'"
The momentum of the system before the collision is equal to the momentum of the system after the collision. In this case, we have the bullet with mass 12.0 g and an unknown velocity, hitting the wooden block at rest, which has a mass of 1100 g.
After the collision, the combined mass (the bullet and the block) swings up to a height of 8.50 cm. This is like a little leap of joy for the bullet and the block!
We can use the conservation of momentum to find the velocity of the combined mass immediately after the collision. We can also use the concept of conservation of energy to find the kinetic energy at that moment. It's like the double feature of physics!
So, let's dive in deep. The momentum before the collision is given by the formula p = mv, where p is momentum, m is mass, and v is velocity. The momentum after the collision is also given by p = mv, where m and v are the combined mass and velocity after the collision.
Since momentum is conserved, we can set the before and after momenta equal to each other: (12.0 g)(v bullet) = (1100 g + 12.0 g)(v combined)
To find the velocity of the combined mass, we need to find the velocity of the bullet. We can do this by using the conservation of energy. Energy is like a magic trick – it can change form, but it doesn't disappear.
The initial kinetic energy of the bullet is given by the formula KE = (1/2)mv^2, where KE is kinetic energy, m is mass, and v is velocity. We know the mass is 12.0 g and the velocity is unknown. The final kinetic energy is zero, as the bullet gets stuck in the wooden block and stops moving.
By setting the initial kinetic energy equal to zero, we can solve for v bullet: (1/2)(12.0 g)(v bullet)^2 = 0
Well, what a coincidence! This tells us that v bullet = 0. You know what they say, "Bullet? More like a Bull-not-moving!"
Now that we've found the velocity of the bullet, we can substitute it back into the momentum conservation equation to find the velocity of the combined mass: (12.0 g)(0) = (1100 g + 12.0 g)(v combined)
This simplifies to 0 = (1112.0 g)(v combined)
Now, we can solve for v combined: v combined = 0
Wait a minute, did I just do that right? Oh, "deer"! It seems like I made a mistake! You'd have to go back and check my calculations because this answer just doesn't make any sense!
I apologize for the confusion; it looks like I gave you a clowntastrophic answer. But don't worry! I always bounce back like a rubber ball! Let's try solving the problem again to ensure we get the accurate answer.
To find the kinetic energy of the combined mass immediately after the collision, we can use the principle of conservation of energy.
The initial kinetic energy (K_i) of the bullet can be calculated using the formula:
K_i = (1/2) * m * v^2
Where:
m is the mass of the bullet
v is the velocity of the bullet
Given:
Mass of the bullet (m) = 12.0 g = 0.012 kg
We need to find the velocity of the bullet (v). To do this, we can use the conservation of momentum:
m_b * v_b = (m_b + m_w) * v
Where:
m_b is the mass of the bullet
v_b is the velocity of the bullet before the collision
m_w is the mass of the wood block
v is the velocity of the combined mass after the collision
Given:
Mass of the wood block (m_w) = 1100 g = 1.1 kg
Rearranging the equation, we can solve for v_b:
v_b = (m_b + m_w) * v / m_b
Now, we can substitute the given values:
v_b = (0.012 + 1.1) * v / 0.012
Next, we can use the conservation of momentum to find the velocity of the combined mass after the collision:
m_b * v_b + 0 = (m_b + m_w) * v
Rearranging the equation and substituting the values:
v = m_b * v_b / (m_b + m_w)
Now we have the value of v, we can calculate the kinetic energy of the combined mass using the formula:
K_f = (1/2) * (m_b + m_w) * v^2
Substituting the values:
K_f = (1/2) * (0.012 + 1.1) * v^2
Finally, we can calculate the final kinetic energy:
K_f = (1/2) * (1.112) * v^2
To find the kinetic energy of the combined mass immediately after the collision, we need to use the principle of conservation of energy.
The principle of conservation of energy states that the total mechanical energy of a system remains constant if no external forces are acting on it. In this case, we can assume that there are no significant external forces, such as friction or air resistance, acting on the bullet-block system.
We can break down the total mechanical energy into two components:
1. Kinetic energy before the collision.
2. Potential energy after the collision.
Let's analyze each component separately:
1. Kinetic energy before the collision:
The kinetic energy of the bullet before the collision can be calculated using the formula:
Kinetic energy = (1/2) * mass of the bullet * velocity^2
Since the mass of the bullet is given as 12.0 g and kinetic energy is directly proportional to the square of the velocity, we don't need the value of the velocity to find the kinetic energy at this stage.
2. Potential energy after the collision:
After the collision, the combined mass (bullet + block) swings up to a height of 8.50 cm. At this height, the potential energy is given by:
Potential energy = (mass of the combined system) * g * height
Here, the mass of the combined system is the sum of the bullet mass (12.0 g) and the block mass (1,100 g), and g is the acceleration due to gravity (approximately 9.8 m/s^2). We need to convert the height from cm to meters to maintain consistent units.
Once we calculate both the kinetic energy before the collision and the potential energy after the collision, we can add them together to find the total mechanical energy of the system immediately after the collision.
Please note that in this calculation, we assume an ideal scenario without accounting for any energy losses due to factors like heat, sound, or deformations.
To summarize:
1. Calculate the kinetic energy of the bullet before the collision using the formula mentioned above.
2. Calculate the potential energy of the combined system after the collision using the formula mentioned above.
3. Add the kinetic energy and potential energy to find the total mechanical energy of the system immediately after the collision.