A 2.50g bullet, traveling at a speed of 425m/s, strikes the wooden block of a ballistic pendulum. The block has a mass of 270g

(a) find the speed of the bullet/block combination immediately after the collision
(b) how high does the combination rise above its initial position?

(a) (bullet mass) x (bullet initial velocity) = (total mass) x (speed of bullet & block)
Solve for (speed of bullet & block)

(b) since kinetic energy of bullet & block = increase in potential energy at top of swing
g h = (1/2) (speed of bullet & block)^2

A 2.50-g bullet, traveling at a speed of 425 m/s. strikes the wooden block of a ballistic pendulum, such as that in Figure 7.12. The block has a mass of 215 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?

(a) To find the speed of the bullet/block combination immediately after the collision, we can use the conservation of momentum:

(bullet mass) x (bullet initial velocity) = (total mass) x (speed of bullet & block)

Substituting the given values:
(2.50g) x (425m/s) = (2.50g + 270g) x (speed of bullet & block)

Dividing both sides by (2.50g + 270g):
(2.50g x 425m/s) / (2.50g + 270g) = (speed of bullet & block)

Simplifying the equation:
(2.50 x 10^-3 kg x 425m/s) / (2.50 x 10^-3 kg + 270 x 10^-3 kg) = (speed of bullet & block)

Calculating the result:
(1.0625 kg*m/s) / (0.2725 kg) = (speed of bullet & block)

Therefore, the speed of the bullet/block combination immediately after the collision is approximately 3.90 m/s.

(b) To find the height the combination rises above its initial position, we can use the conservation of mechanical energy:

Kinetic energy of bullet & block = Increase in potential energy at the top of the swing

Using the equation:
g h = (1/2) (speed of bullet & block)^2

Substituting the known values:
(9.8 m/s^2) x h = (1/2) (3.90 m/s)^2

Simplifying the equation:
9.8 m/s^2 x h = 0.5 x (3.90 m/s)^2

Calculating the result:
9.8 m/s^2 x h = 0.5 x 15.21 m^2/s^2

Therefore, the height the combination rises above its initial position is approximately 0.785 meters or 78.5 centimeters.

To solve this problem, we need to use the principles of conservation of momentum and conservation of energy. Let's go step by step to find the answers to both parts of the question:

(a) To find the speed of the bullet/block combination immediately after the collision, we can use the conservation of momentum principle. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.

The formula for momentum is given by mass times velocity (p = mv).

Given values:
Bullet mass (m1) = 2.50g = 0.0025kg
Bullet initial velocity (v1) = 425m/s
Block mass (m2) = 270g = 0.27kg

Total initial momentum before the collision (p_initial) = (m1 * v1)

Let the speed of bullet/block combination after the collision be v2.

Total momentum after the collision (p_final) = (m1 + m2) * v2

According to the conservation of momentum principle, p_initial = p_final.

So, (m1 * v1) = (m1 + m2) * v2

We can rearrange the equation to solve for v2:

v2 = (m1 * v1) / (m1 + m2)

Now we substitute the known values:

v2 = (0.0025kg * 425m/s) / (0.0025kg + 0.27kg)

Calculate the value of v2 to get the speed of the bullet/block combination immediately after the collision.

(b) To find how high the combination rises above its initial position, we can use the conservation of energy principle. The increase in potential energy at the top of the swing is equal to the kinetic energy of the bullet/block combination just after the collision.

The formula for kinetic energy is given by (KE = 1/2 * mv^2) and the formula for potential energy is given by (PE = mgh).

In this case, as the bullet/block combination rises, it converts all its kinetic energy into potential energy at the top of the swing.

So, we can equate the kinetic energy of the bullet/block combination just after the collision with the potential energy at the top of the swing:

(1/2 * (m1 + m2) * v2^2) = (m1 + m2) * g * h

Now we can rearrange the equation to solve for h:

h = (1/2 * v2^2) / g

Substitute the known values:

h = (1/2 * (v2)^2) / g

Calculate the value of h to find out how high the combination rises above its initial position.

Remember to convert the mass given in grams to kilograms before substituting it into the equations.

I hope this explanation helps you understand how to solve the problem.

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