1. Under what conditions is it necessary to use numerical methods to solve physics problems?[br]2. What function [math]r_x(t)[/math] solves the equation [math]\frac{d^2r_x}{dt^2}=g[/math] where [math]r_x(t=0)=1.0m[/math] and [math]v_x(t=0)=2.0m/s[/math], where g is the gravitational field strength near earth's surface?[br]3. Using [math]k=\frac{1}{2}\rho Ac_d[/math] to account for all constants in the drag equation, how fast will a 1kg mass fall at terminal speed given k=0.25?  How fast (terminal) will a 2kg mass fall with the same k?  Please do this problem both by hand and using numerical methods.[br]4. In the problem above, how long does it take for the 1kg mass dropped from rest to reach 90% of terminal speed?  What about for the 2kg mass? (This will require numerical methods)[br]5. Given the rocket in the text, how high will it fly if we allow its thrust to continue for 20s?  What maximum speed upward and then falling downward will it reach before hitting the ground? It would be a good idea to create your own model using numerical methods to answer this problem. We will have similar such problems on our first numerical methods exam.[br]6. If the electric car example in the last section was accelerating up a 2% grade, what would its 0-60mph time, quarter mile time and top speed be?  What about for a 2% downward grade?  Recall that you may use a small angle approximation for such small grades.[br]7. If an electric car has a motor force function given by [math]F(v)=5000-60v[/math] and we want the car to have twice the force with which to accelerate (at the expense of speed) by changing the gearing, what would the new force function be? Hint: The product of max force and max speed as defined by the y and x intercepts respectively must remain constant. [br]8. For a car to increase it's cruising speed by 25% when air drag is the dominating resistive force, by what factor must power be increased? By the way, for bicycles going over 20mph and cars around highway speed or above, drag IS the dominant resistive force.

1. Please read opening section.[br]2. x(t)=At[sup]2[/sup]+Bt+C; A=g/2, B=2.0m/s, C=1.0m[br]3. 6.26m/s; 8.85m/s[br]4. 0.93s; 1.32s[br]5. 2223m; 130m/s; -135m/s[br]6. 7.2s; 15.4s; 59.3m/s; 6.5s; 14.7s; 63.3m/s[br]7. F(v)=10000-240v[br]8. 1.95... or nearly double!