In a ballistic pendulum experiment, projectile 1 results in a maximum height h of the pendulum equal to 2.2 cm. a second projectile causes the the pendulum to swing higher, to h2 = 5.4 cm. the second projectile was how many times faster than the first?

The height is proportional to the kinetic energy of the projectile. So the ratio of the two KEs is 5.4/2.2 = 2.45.

KE is proportional to velocity squared, so velocity is proportional to the square root of the KE, which is √2.45 = 1.57, which is the answer. 

To explain it more in detailed: 

KE = PE = mgh, which displays you the KE and height proportionality. 

KE = ½mV² or 

V = √(2KE/m) which shows you the other proportionality 

combining 

V = √(2mgh/m) = √(2gh) 

V₁/V₂ = √(2gh₁) / √(2gh₂) = √(h₁/h₂) = √(5.4/2.2) = 1.57