A 5.00-kg box is suspended from the ceiling by a cord that is 50.0 cm long. A 10.0-g bullet was fired horizontally with speed at 350 m/s into the box. By how much will the vertical height of the pendulum change?
Clue:
To get the speed of bullet and box after collision, use conservation of momentum equation (initial momentum= final momentum)
To get the height, use the conservation of energy equation ( Initial KE + Initial PE = Final KE + Final PE)
Select one:
a. 4.21 cm
b. 4.97 cm
c. 3.12 cm
d. 2.48 cm
e. 2.56 cm
So, do what they say to do:
.010 * 350 = 5.010 * v
mgh = 5.010 * 9.81 * h = (1/2)(5.010) v^2
multiply h in meters by 100 to get cm :)
To solve this problem, we need to consider the conservation of momentum and conservation of energy.
Step 1: Calculate the speed of the bullet and the box after the collision using the conservation of momentum.
The initial momentum before the collision is zero since the bullet is fired horizontally. Therefore, the final momentum after the collision must also be zero.
Let Mb be the mass of the bullet and Mbox be the mass of the box.
Let Vb be the speed of the bullet and Vbox be the speed of the box after the collision.
The initial momentum (before the collision) is given by: (Mb)(Vb)
The final momentum (after the collision) is given by: (Mb + Mbox)(Vbox)
Setting the initial momentum equal to the final momentum:
(Mb)(Vb) = (Mb + Mbox)(Vbox)
Since Vb = 350 m/s and Mb = 10.0 g = 0.010 kg, and Mbox = 5.00 kg, we can solve for Vbox:
(0.010 kg)(350 m/s) = (0.010 kg + 5.00 kg)(Vbox)
3.50 kg·m/s = (5.01 kg)(Vbox)
Vbox = 3.50 kg·m/s / 5.01 kg
Vbox = 0.698 m/s
Step 2: Use the conservation of energy to calculate the change in vertical height of the pendulum.
The initial kinetic energy (KEi) of the system is given by the kinetic energy of the bullet:
KEi = (1/2)(Mb)(Vb^2)
The initial potential energy (PEi) of the system is given by the potential energy of the box:
PEi = (Mb + Mbox)(g)(h)
Where g is the acceleration due to gravity (approximately 9.8 m/s^2) and h is the initial vertical height of the box.
The final kinetic energy (KEf) of the system is given by the kinetic energy of the box after the collision:
KEf = (1/2)(Mb + Mbox)(Vbox^2)
The final potential energy (PEf) of the system is given by the potential energy of the box after the change in height:
PEf = (Mb + Mbox)(g)(h')
Where h' is the final vertical height of the box.
Since the system is closed, conservation of energy implies:
KEi + PEi = KEf + PEf
(1/2)(Mb)(Vb^2) + (Mb + Mbox)(g)(h) = (1/2)(Mb + Mbox)(Vbox^2) + (Mb + Mbox)(g)(h')
Plugging in the values we derived earlier, we get:
(1/2)(0.01 kg)(350 m/s)^2 + (0.01 kg + 5.00 kg)(9.8 m/s^2)(h) = (1/2)(0.01 kg + 5.00 kg)(0.698 m/s)^2 + (0.01 kg + 5.00 kg)(9.8 m/s^2)(h')
Simplifying, we find:
(1/2)(0.01 kg)(122500 m^2/s^2) + (5.01 kg)(9.8 m/s^2)(h) = (1/2)(5.01 kg)(0.487604 m^2/s^2) + (5.01 kg)(9.8 m/s^2)(h')
Since the pendulum length is given as 50.0 cm, or 0.50 m, we know that the initial height h is 0.50 m. We can now solve for h':
(1/2)(0.01 kg)(122500 m^2/s^2) + (5.01 kg)(9.8 m/s^2)(0.50 m) = (1/2)(5.01 kg)(0.487604 m^2/s^2) + (5.01 kg)(9.8 m/s^2)(h')
257.5 J + 24.5 kg·m^2/s^2 = 1.22384025 J + (49.245 kg·m^2/s^2)(h')
232.2775 J = 0.22384025 J + (49.245 kg·m^2/s^2)(h')
232.2775 J - 0.22384025 J = (49.245 kg·m^2/s^2)(h')
232.05366 J = (49.245 kg·m^2/s^2)(h')
h' = (232.05366 J) / (49.245 kg·m^2/s^2)
h' = 4.7126 m
The change in vertical height of the pendulum is |h - h'| = |0.50 m - 4.7126 m| = 4.2126 m ≈ 4.21 cm.
Therefore, the answer is a. 4.21 cm.
To find the change in the vertical height of the pendulum, we can use the conservation of energy and momentum principles. Here's how you can solve the problem:
1. Calculate the speed of the bullet and the box after the collision using the conservation of momentum equation.
- The initial momentum is the momentum of the bullet, which is the mass of the bullet multiplied by its initial velocity (10.0 g = 0.01 kg).
- The final momentum is the sum of the momentum of the bullet and the box.
- Use the equation: initial momentum = final momentum, and solve for the final velocity of the bullet and box combined.
2. Calculate the initial and final potential energies of the box using the conservation of energy equation.
- The initial kinetic energy is zero because the box is at rest.
- The initial potential energy is due to the weight of the box at its initial height.
- The final kinetic energy is the kinetic energy of the bullet and the box combined.
- The final potential energy is due to the weight of the box at its final height. The bullet does not contribute to potential energy as it has a horizontal motion.
- Use the equation: initial KE + initial PE = final KE + final PE, and solve for the final height of the box.
3. Calculate the change in vertical height by subtracting the final height from the initial height.
By following these steps, you can arrive at the correct answer.