This section is dedicated to teachers. In case that students are also interested they can do it as optional work !

According to the theory the maximale distance will be reached with an angle α = 45° . If you put this together with the normaly by good athlets reached velocity v₀ = 9 ms^-1 in the mathematical model, you will get a maximal jump distance of x_max = 8.3 m, a realistic value (world record is 8.95 m). But the maximal jump heigth y₀ by the model is 2.1 m, what is never reached in praxis. What is incorrect in our model ?

First, the landing point is under the level of the point of taking-off. The relevant point is allways the gravity center of the body. In case of jumping off this point is on the level of the hips of the jumper, in case of landing on the ground level. This heigth difference is about 1 m. In that case the optimal angle for jumping α is 42°.
Second, the jumper as usual doesn't reach the necessary vertical velocity, because running he reaches only horizonzal velocity. Good jumpers reduce their horizontal velocity before they jump (with a special technique of running for the last three steps). They can reach 10 ms^-1 , reduce suddenly the horizontal velocity to 8.5 ms^-1 and reach a horizontal velocity of 3 ms^-1 This gives a jump angle of tan α =3/8.5 ➝ α = 19.4°, a second reason for the loss of max width.