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

When the spring is compressed a distance \( d \), it stores potential energy given by the formula \( PE = \frac{1}{2} k d^2 \), where \( k \) is the spring constant. Launching a mass \( m \) converts this energy into gravitational potential energy \( PE = mgh \). For the first situation, we can set the energies equal to each other and solve for \( h \). Now, when the spring is compressed to \( 2d \), the energy stored becomes \( \frac{1}{2} k (2d)^2 = 2 k d^2 \). Launching a mass \( 2m \), we set \( 2 k d^2 = 2mgh' \), leading us to find that \( h' = 2h \). So, the second mass rises to \( 2h \). Now imagine you're at a carnival, and instead of a boring physics lesson, you get to launch a rubber chicken! The mechanics are the same, but instead of worrying about mass and height, you're trying to send your rubber chicken soaring to the highest point. Compress the spring further, and that cheeky chicken goes even higher! Physics is just a game when you think about it that way!