A 3.67-g bullet, traveling at a speed of 484 m/s, strikes the wooden block of a ballistic pendulum, such as that in Figure 7.14. The block has a mass of 186 g. (a) Find the speed of the bullet/block combination immediately after the collision. (b) How high does the combination rise above its initial position?
To find the speed of the bullet/block combination immediately after the collision, we can use the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum of a system remains constant if no external forces act on it. In this case, we can consider the bullet and the block as a closed system.
The momentum of an object can be calculated by multiplying its mass by its velocity. So, the initial momentum of the bullet is given by:
Initial momentum of the bullet = mass of the bullet * initial velocity of the bullet
Since the bullet has a mass of 3.67 g (0.00367 kg) and an initial velocity of 484 m/s, we can calculate its initial momentum:
Initial momentum of the bullet = 0.00367 kg * 484 m/s
Next, we need to consider the block and its velocity before the collision. Since the block is initially at rest, its initial momentum is zero.
According to the principle of conservation of momentum, the total momentum after the collision must be equal to the total momentum before the collision. Therefore:
Total momentum before collision = Total momentum after collision
Initial momentum of the bullet = Final momentum of the bullet + Final momentum of the block
Since the bullet and the block stick together after the collision, their final momentum is equal. Let's call their common final velocity "v".
Final momentum of the bullet = (mass of the bullet + mass of the block) * v
Now, we can set up the equation and solve for the final velocity:
0.00367 kg * 484 m/s = (0.00367 kg + 0.186 kg) * v
Rearranging the equation, we get:
v = (0.00367 kg * 484 m/s) / (0.00367 kg + 0.186 kg)
Once you substitute the values and calculate the final velocity, you will have your answer to part (a).
To determine how high the combination rises above its initial position, we can use the principle of conservation of mechanical energy.
The principle of conservation of mechanical energy states that the total mechanical energy of a system remains constant if no external non-conservative forces (like friction) act on it.
The mechanical energy of the system is the sum of the kinetic energy (KE) and the potential energy (PE) at any given point.
Initially, the combination is at rest, so its initial kinetic energy is zero. The potential energy is given by the height of the system from its initial position.
When the combination reaches its maximum height, all of its initial kinetic energy is converted into potential energy.
Using the conservation of mechanical energy, we can equate the initial kinetic energy to the potential energy at maximum height.
Initial kinetic energy = Potential energy at maximum height
0.5 * (mass of the bullet + mass of the block) * (final velocity)^2 = (mass of the bullet + mass of the block) * g * h
Here, g is the acceleration due to gravity (approximately 9.8 m/s^2) and h is the maximum height we want to find.
Rearranging the equation, we get:
h = (0.5 * (mass of the bullet + mass of the block) * (final velocity)^2) / ((mass of the bullet + mass of the block) * g)
Once you substitute the values and calculate the maximum height, you will have your answer to part (b).