Angular momentum vector of mass segment - Mass segment located at r T. with velocity v (a) Image credit: OpenStax University Physics Physics simulation is a very important branch of computer science, with applications to computer game design, computer graphics, and various scientific fields. At its most basic, a physics simulation needs to understand the net force on an object or particle, which by Newton's second law gives the acceleration, and use this to compute the object's velocity and position over time. Many sophisticated methods have been developed to accurately predict the positions of objects based on the forces applied, such as the classic Runge- Kutta method (RK4). We will not be using any of these sophisticated methods. Instead, we will be using a form of Euler's method, the simplest and (often) least accurate such method, to compute velocity, and using the trapezoidal rule to compute position. The scenario we will be attempting to model is that of a puck sliding down a hill. If we ignore all forces except gravity and the normal force from the hill, then the equation that gives us the acceleration of the puck is:' -gVf ma = 1+ 1/Vf1 12 where m is the mass, a is the acceleration as a 2D vector, g is the magnitude of gravity, and f is the height of the hill as a function of the 2D position. The hill in question is defined by: f(x. y) = -12 - y? where r and y are in meters. We will take g = 10m/s' and m = 1kg. 1 If the notation ||Vf||2 is unfamiliar to you, it is simply the dot product Vf - Vf, i.e.(a) (2 points) At time to = 0 seconds, the puck is at position (Co. yo) = (1, 0) ( meters). What is the gradient Vf of the slope at this point? (b) (1 point) Based on the given force equation, solve for the acceleration a. (c) (1 point) The initial velocity of the puck is (up, vo) = (0, 1) in m/s. Euler's method is to approximate the velocity (un, 1 ) at time t1 = 1 by using the following equation: (21, ") = (10, UD) + (t1 - to)a (This is effectively assuming acceleration remains constant throughout the second.) Using this approximation, compute the new velocity (#1, v1).(d) (1 point) To apply the trapezoidal rule and estimate the position (21, y1), we will take the average of the velocity at the start and the end of the time interval. The equation for this approximation is: 3 (21, 31) = (10. 30) + (t1 - to) -20: 20) + (21, v1 ) 2 Use this approximation to compute the new position (r1, y1).(e) (2 points) These approximation methods work best when we perform multi- ple steps. We will not perform an entire second step, but we should at least compute the new acceleration (among other things, this will tell us how accurate our assumption that acceleration would remain relatively constant was). So what is the gradient Vf of the slope at (21, (1 )? (f) (2 points) As with last time, find the acceleration at (21, y1) from the gradient. Is this close to the previous acceleration