Physics in Gymnastics

            I used to love the show Make it or Break it, which follows the lives of a few gymnasts. I was always impressed by the moves and jumps that the gymnasts were able to do. I found the vault to be one of the most impressive events. In the video above, one of the characters from the show, Kaylie, preforms the vault. To perform a vault, a gymnast begins by running down a runway, which is usually padded or carpeted. Then, they jump onto a springboard and spring onto the vault with their hands. In the video, Kaylie is doing a vault from the Yurchenko family, in which a gymnast puts their hands onto a mat that is placed before the springboard, does a round-off onto the board and then does a back handspring onto the vault. After springing onto the vault, the gymnast proceeds with their off-flight, which can be as simple as leaping over the vault or as complicated as doing several twists and turns in the air. The gymnast then lands on a mat on the other side of the vault. 

According to Newton’s second law, the more mass and acceleration a gymnast has when running towards the vault, the greater the force they exert on the springboard will be. Gymnasts must reach a maximum velocity while running to the springboard in order to produce a maximum force. If we assume that Kaylie’s mass is 45 kg and that she is running to the spring board with a velocity of 4 m/s over a period of 5 seconds, what would the force that she exerts be?

F=ma

F=m(v/t)

F=(45 kg)((4 m/s)/(5 s))

F=36 N

Newton’s third law states that for every action, there is an equal but opposite reaction. Therefore, when a gymnast jumps on a springboard, she exerts a force on the springboard, which exerts an equal force back on the gymnast, which then propels her into the air. The angle at which the gymnast hits the springboard is very important because the more vertical the angle is, the greater the force will be to propel the gymnast higher into the air. Let’s say that Kaylie jumps onto the springboard at an angle of 80 degrees, which is not completely vertical. What would the new force that she is exerting be?

F_f = F_i sin Θ

F_f = (36 N)sin(80^o)

F_f = 35.45 N (less than the original force of 36 N)

  

When a gymnast then pushes off the vault, her hands exert a downward force on the vault and the vault exerts an equal force on her. Due to the conservation of momentum, gymnasts cannot gain any angular momentum once they spring off the vault. The greater the angular momentum, the more potential the gymnast has for doing flips in the air. Gymnasts can gain angular momentum by pushing off from the vault at an angle. Once they are in the air, gymnasts often curl into a tight ball to achieve more flips because the smaller their radius, the faster they spin. When they are about to land, gymnasts then extend their bodies because that slows down their rotational speed, which ensures that they will land smoothly on their feet.

We can calculate the initial energy that a gymnast has during her run to the vault, which is the maximum energy she will have throughout the entire event. A gymnast's initial energy is kinetic energy. The greater the velocity is that the gymnast has while running, the greater the kinetic energy is that she will have to complete the rest of the vault. What is Kaylie’s kinetic energy at the beginning of her run?

            KE= ½ mv²

            KE= ½(45 kg)(4 m/s)²

            KE= 360 J

           After the run, the gymnast jumps onto the springboard and her kinetic energy is then transferred into the springs of the board as spring potential energy. Assuming that no energy is lost due to friction or air resistance, the potential energy would be the same as the kinetic energy. If we assume that the spring compresses 0.15 meters when Kaylie jumps onto the springboard, what would the spring constant be?

           PE_s= ½ kx²

           360 J= ½ k(0.15 m)²

           k= 32,000 N/m

As the gymnast pushes off the vault, the spring potential energy is then converted to kinetic energy and gravitational potential energy. As the gymnast flips through the air, her kinetic energy increases and her potential gravitational energy decreases. However, throughout the entire vault, energy is conserved.