Partnership for Integration of Computation into Undergraduate Physics

## Exercise 1# In these exercises we model the snowboarder during the landing by having them bend their knees as they touch down. During the landing their center of mass drops through a vertical distance $b$ (thus $b$ is along the $y'$ axis). $Axes\ note$: In the exercises below, the unprimed coordinates $(x, y)$ are rotated so that the $x-axis$ points down the slope. The primed coordinates $(x', y')$ are horizontal/vertical. a) A snowboarder drops from a height $h$ landing on a sloping surface. Show that the speed of the snowboarder perpendicular to the slope (i.e. along $y$) when they first touch the surface is: $$v_y = -\sqrt{2gh}\ \cos\theta.$$  Figure 1. A snowboarder drops (with zero initial speed) off a ledge, landing on a sloping snow surface. b) During the landing the snowboarder drops their center of mass vertically by an amount $b$ by bending their knees. After their center of mass has dropped by this amount their velocity along the $y$ direction is zero. Show that their average acceleration perpendicular to the slope (i.e. along $y$) during this landing is: $$a_N = \frac{gh\cos\theta}{b}.$$ c) Use Newton's second law to show that the average normal force exerted by the slope on the snowboarder, i.e. the force the snowboarder feels during landing, is given by the expression below. Assume friction between the snowboard and the surface can be neglected. $$F_N=mg\left(1+\frac{h}{b}\right)\cos\theta.$$ d) Find an expression for the duration time for the landing (from when the snowboarder first touches the ground to when their speed along $y$ is zero). e) Discuss any assumptions or approximations made in the model of this exercise and what effect they might have on the force on landing. ## Exercise 2 # The python program 'Snowboarder-1' calculates the average acceleration $a_N$ (along $y$) and the average force during landing $F_N$ (along $y$) as a function of the ledge height $h$ and the slope angle $\theta$. a) Plot $a_N$ and $F_N$ as a function of slope angle $\theta$ $(0\leq\theta < 90^o)$ for the case of a drop height $h$ of 3.0 m. Assume a snowboarder mass of 72 kg and that their center of mass drops a distance of 0.40 m during landing. What is the value of $F_N$ for a person who drops onto horizontal ground? b) For the case of part a), what is the value of $F_N$ for a person who drops onto a slope $\theta$ of $60^o$? c) Assume that the force during landing in part b) yields a comfortable landing for the snowboarder. What slope (value of $\theta$ ) must the snowboarder land on if the landing from a 5.0 m height is comfortable? While the calculated slope turns out to be steep, snowboarders experience such slopes in the sport of extreme snowboarding. A web search of the phrase “Extreme Snowboarding Youtube” or the specific site “https://www.youtube.com/watch?v=5xbb9N8PpN8” yield examples. d) Look at your plots of average force and acceleration during the landing versus $\theta$. Infer from these plots what parameters in Newton's second law are changing with $\theta$ to allow the force to be small at large values of $\theta$. No calculation required here- just a discussion of why the landing force is small at larger $\theta$. e) If the snowboarder lands in 4.0 cm of soft fresh power how might you include this in your calculation of part a)? What are the new values of $F_N$ in a) and b)? ## Exercise 3# We now consider a snowboard jump where the snowboarder requires some horizontal speed to reach the slope as shown below. This shape is somewhat idealized and we discuss this further at the end of the exercise. We'll do a calculation of the force the snowboarder experiences during landing provided they land on the slope at the lip.  Figure 2. A snowboarder travels off a ledge with an initial speed $v_{ox}'$ that just allows them to reach the lip of the slope. They land on a sloping snow surface. a) Show that in order for the snowboarder to land on the slope at the lip, the horizontal launch speed must be $v_{ox}'=d\sqrt{g/(2h)}$ b) Hence show that the landing speed and the angle of the landing velocity $\phi$ (measured below the horizontal) are $$v = \sqrt {g\left( \frac{d^2}{2h}+2h \right) }.$$ $$\tan\phi=\frac{2h}{d}.$$ c) During the landing the snowboarder drops their center of mass vertically by an amount $b$ by bending their knees. Show that their average acceleration perpendicular to the slope (i.e. along $y$) during this landing is: $$a_y\equiv a_N = g\left( \frac{d^2}{2h}+2h \right)\frac{\sin^2(\phi -\theta)}{2b\cos\theta}.$$ d) Assume friction between the snowboard and the surface can be neglected. Use Newton's second law to show that the average normal force, i.e. the force the snowboarder feels during landing, is:$$F_y \equiv F_N=mg\left(1+\frac{(2h-d\tan \theta)^2}{4hd} \right)\cos\theta.$$ e) Find an expression for the duration time for the landing (from when the snowboarder first touches the ground to when their speed along $y$ is zero). f) In the limit where the landing angle $\theta \rightarrow \phi$ from below, what is the limiting value of the landing force? Explain how to interpret the formula for duration time of the landing in e) in this limit. g) The model discussed here is a more general case than that discussed in Exercise 1. What values of the parameters in this model will yield the special case of Exercise 1? Now check each of the results above (landing speed $v$, landing angle $\phi$, acceleration $a_N$, force during landing $F_N$, and time duration of landing $\Delta t$) in this limit. Make sure you obtain the same results as in Exercise 1. h) Discuss any assumptions or approximations made in the model of this exercise, Figure 2. What effect do these assumptions have on the force on landing? In particular discuss what assumptions were made in choosing the shape of the jump, and how general the shape of the jump is. ## Exercise 4 # a) Write a program (we'll call it "Snowboarder-2") analyzing the situation of Exercise 3 using the python program 'Snowboarder-1' as a starting point. This program should: * calculate the launch speed $v_{ox}'$ for the snowboarder to reach the lip for a particular value of height $h$ and length $d$ of the jump * calculate the resulting landing speed $v$ and the landing angle, $\phi$ (measured from the horizontal) * plot the acceleration during landing perpendicular to the slope, $a_N$ as a function of the slope angle, $\theta$ (plotted from $0^0\ to\ \phi$) * plot the force during landing perpendicular to the slope, $F_N$ (force exerted by the slope on the snowboarder) as a function of the slope angle, $\theta$ (plotted from $0^0\ to\ \phi$) * plot the duration time of the landing as a function of the slope angle, $\theta$ (plotted from $0^0\ to\ \phi$) b) Plot $a_N$ and $F_N$ as a function of slope angle $\theta$ $(0\leq\theta < \phi)$ for the case where $h$ is 3.0 m and $d$ is 2.0 m. Assume a snowboarder mass of 72 kg and that their center of mass drops a distance of 0.40 m during landing. What is the value of $F_N$ for a person who drops onto a slope inclined at $60^0$? How does this compare to the force calculated in Exercises 2b)? c) Consider somewhat larger values of $h$ and $d$. Take the case $h$ is 10 m, $d$ is 20 m ($b$ is still 0.4 m). What is the required take-off speed to reach the lip? What is the landing speed? Does the snowboarder land at the lip if the slope $\theta$ is $60^o$? Hint: think about the relative values of $\theta$ and $\phi$. d) Look at your plots of $F_N$ versus $\theta$ from part b) and part c). At larger values of $\theta$ the landing force becomes small. Discuss how the model of this exercise is able to further reduce the force on landing as compared to the model of Exercises 1 and 2. Include discussion of the special case of a landing corresponding to $\theta = \phi$. e) It is not possible for a snowboarder to always have the correct take-off speed. How might you modify the shape of the jump to make it safer for the snowboarder? Explain your answer(s). f) Finally discuss how you might modify the calculation of 4b) if the snowboarder lands in soft snow that is 6 vertical centimeters deep on the slope. g) Maximum forces on the ankle joint have been measured during a jumping motion [see Clin Biomech (Bristol, Avon). 2013 Jan; 28(1): 98–103. Daniel J Cleather, Jon E Goodwin, and Anthony MJ Bull* or see link https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3966561/] and was found to be in the range $8.9-10.0 mg$ over a short time. Assuming the snowboarder does not land awkwardly we take this as a good estimate of the maximum possible force on landing the snowboarder can land safely with. Look back at your answers for the force on landing on a slope ($\theta > 0^0$), exercises 2 and 4, and comment about if this limit is exceeded. Now look back at any of your calculations for landing on horizontal ground ($\theta = 0^0$) and again comment on if this limit is exceeded. *We thank an anonymous reviewer for pointing out this reference. ## Exercise 5 # Consider what happens when the slope angle $\theta$ is greater than the landing angle at the lip. We assume the snowboarder still has the right speed to reach the lip, namely $v_{ox}'=d\sqrt{g/(2h)}$. In this case the snowboarder skims over the lip and lands further down the slope - the designer of this snowboard jump has done a poor job!  Figure 3. A snowboarder travels off a ledge with an initial speed $v_{ox}'$ that just allows them to reach the lip of the slope. They skim over the lip and land further down the slope at position $(x_1', y_1')$. a) In the $x' - y'$ coordinate system show that the snowboarder lands at $$x_1'= \frac{d^2}{h}\tan\theta-d,\quad\quad\quad y_1'=\left(2-\frac{d}{h}\tan\theta\right)d\tan\theta$$ b) Hence prove that the landing speed and landing angle are: $$v_1=\sqrt{g\left(\frac{d^2}{2h}+2\left(\frac{d}{h}\tan \theta -2\right)d\tan \theta +2h\right)},\quad\quad \tan\phi_1=2\sqrt{\left(\tan \theta -\frac{2h}{d}\right)\tan \theta+\frac{h^2}{d^2}}$$ c) During the landing the snowboarder drops their center of mass vertically by an amount $b$ by bending their knees. Show that the acceleration and force perpendicular to the slope (i.e. along the $y$ - axis) during landing are: $$a_y\equiv a_N=\frac{v_1^2\sin^2(\phi_1-\theta)}{2b\cos\theta}, \quad\quad F_y\equiv F_N=m\left(g\cos\theta+\frac{v_1^2\sin^2(\phi_1-\theta)}{2b\cos\theta}\right).$$ d) Check that if $\theta = \phi_1$ then the force calculated in 5c) reduces to the expected value. e) Find an expression for the duration time, $\Delta t$, of the landing (i.e. time from when the snowboarder first touches the ground to when their speed along $y$ is zero). # Exercise 6 a) Use the python program you wrote for Exercise 4 as a starting point to write a program (we'll call it 'Snowboarder-3') analyzing the situation of Exercise 5. This program should add to your earlier program by calculating the average acceleration during landing $a_N$ and the average force during landing $F_N$ as a function of the slope angle $\theta$ for the case $\phi$ < $\theta$ < $90^o$. The case $0^o$ $\leq$ $\theta$ < $\phi$) was done in ‘snowboarder-2’ and should be included as part of ‘snowboarder-3’. b) For each of the four cases below, plot $a_N$ and $F_N$ as a function of slope angle $\theta$. If $\theta$ is $60^0$, state (for each of the four cases below) if the snowboarder skims over the lip or lands on the lip and determine the force on landing. * $h$ is 2.0 m and $d$ is 3.0 m, * $h$ is 3.0 m and $d$ is 2.0 m, * $h$ is 10 m and $d$ is 20 m, * $h$ is 20 m and $d$ is 10 m, c) For the case $h$ is $2.0\ m$ and $d$ is $3.0\ m$, check your answer to 5d).