In a ballistic pendulum experiment, projectile 1 results in a maximum height h of the pendulum equal to 2.4 cm. A second projectile causes the pendulum to swing twice as high, h_2 = 4.8 cm. The second projectile was how many times faster than the first?

To determine how many times faster the second projectile was compared to the first, we need to consider the conservation of energy in the system.

In a ballistic pendulum experiment, a projectile is launched into a pendulum and gets embedded into it. The initial kinetic energy of the projectile is converted into potential energy of the pendulum at its maximum height.

Let's use the following variables for our calculation:

h1 = Maximum height of the pendulum caused by the first projectile (2.4 cm)
h2 = Maximum height of the pendulum caused by the second projectile (4.8 cm)
v1 = Velocity of the first projectile
v2 = Velocity of the second projectile

According to the conservation of energy:

Initial kinetic energy of the first projectile (1/2)mv1^2 = Potential energy of the pendulum at height h1, which is mgh1

Initial kinetic energy of the second projectile (1/2)mv2^2 = Potential energy of the pendulum at height h2, which is mgh2

Since the mass (m) of the projectiles is the same and cancels out, we are left with:

(1/2)v1^2 = gh1 ---(equation 1)
(1/2)v2^2 = gh2 ---(equation 2)

To compare the velocities of the two projectiles, we can simplify the equations:

v1^2 = 2gh1
v2^2 = 2gh2

Now, let's find the ratio of v2/v1:

(v2/v1)^2 = (2gh2) / (2gh1)
(v2/v1)^2 = h2 / h1

Given that h2 = 4.8 cm and h1 = 2.4 cm:

(v2/v1)^2 = (4.8 cm) / (2.4 cm)
(v2/v1)^2 = 2

Taking the square root of both sides:

v2/v1 = √2

Thus, the second projectile was √2 (approximately 1.41) times faster than the first projectile.