1. A 125-gram projectile strikes a 1.45 kg block that's attached to a massless vertical pole & becomes embedded in the block causing the bullet & block to swing upward. The overall length of the pole & the block combined is 1.1m. The bullet & block at the highest point makes a 45 degree angle with the original vertical position. What was the intial velocity of the bullet before striking the block?
To find the initial velocity of the bullet before striking the block, we can use the principle of conservation of momentum and the concept of projectile motion.
Step 1: Calculate the momentum before and after the collision.
The momentum before the collision is given by the product of the mass of the bullet (m1) and its velocity (v1), as the block is initially at rest:
Momentum before = m1 * v1
The momentum after the collision is given by the sum of the masses times the velocity of the bullet and the block (v2) after they swing upward:
Momentum after = (m1 + m2) * v2
According to the principle of conservation of momentum, the momentum before and after the collision should be the same:
Momentum before = Momentum after
Step 2: Calculate the velocity of the bullet and block after the collision.
To determine the velocity after the collision, you need to consider the conservation of kinetic energy. At the highest point, all the initial kinetic energy is converted into potential energy.
Using the concept of projectile motion and the given information of the angle (45 degrees) with respect to the vertical position, we can determine the vertical component of velocity (v2y) using the equation:
v2y = v2 * sin(45 degrees)
Since the motion is completely vertical at the highest point, the final velocity in the y-direction is 0. Therefore, we have:
v2y = 0 m/s
Using the vertical component of velocity, we can find the total velocity (v2):
v2 = v2y / sin(45 degrees)
Step 3: Solve for the initial velocity of the bullet.
Substituting the expressions of momentum before and after the collision, we can write the equation as:
m1 * v1 = (m1 + m2) * v2
Rearranging the equation, we can solve for the initial velocity (v1):
v1 = [(m1 + m2) * v2] / m1
Now, substitute the known values (m1 = 0.125 kg, m2 = 1.45 kg, v2 = v2y / sin(45 degrees)) into the equation and solve for v1.