A Hooke's law spring is compressed a distance \( d \) and is used to launch a mass \( m \) vertically to a height \( h \) above its starting position. Under double the compression, the spring is now used to launch the mass. How high does the mass now rise above its starting position? 1) \( 2 h \) 2) \( 1.41 h \) 3) \( 3 h \) 4) \( 4 h \) 5) \( 8 h \) 4 1 \( h \)

When a spring is compressed, the potential energy stored in the spring is given by the formula \( \frac{1}{2} k d^2 \), where \( k \) is the spring constant and \( d \) is the compression distance. When you double the compression to \( 2d \), the potential energy becomes \( \frac{1}{2} k (2d)^2 = 2 k d^2 \), which is four times the original energy (since \( (2d)^2 = 4d^2 \)). This extra energy translates into greater height, so the mass will rise four times as high, or \( 4h \). In terms of real-world applications, this principle has diverse uses, from catapults in ancient warfare to modern-day roller coasters. Engineers and designers utilize Hooke's law to craft systems where springs store and release energy efficiently, such as in vehicle suspension systems or various consumer products like retractable pens and toys. Understanding how compression affects performance can lead to safer and more effective designs!