Welcome to Geneva, home to many rare species of duck. Hello, I'm the daughter of Airfiction. This you teach you for interfering with certain experiments. And how? Someone's shooting me. What did I do? You never did me a life. You bad duck. I catch you and it's how you're big. Just after the bullet is ejected from the gun, the distance between the duck and the bullet is D. The last vector of Bolivie can be expressed is VXI plus VYJ, where I and J unit vectors. We can look at each component separately. VX stays constant at V0X, which is the initial horizontal speed when it was ejected from the gun. VY equals V0Y plus G times T, which is the initial vertical speed for the additional speed caused by acceleration due to gravity. Similarly, the velocity of the duck U is UxI plus UYJ. In this case, Ux equals U0X, which equals 0, because the duck is not... In this case, Ux equals U0X, which equals 0, because the duck is not moving horizontally. Uy equals U0Y plus GT. V0Y and U0Y, both equals 0, that's both objects for with no initial velocity, vertically. Therefore, Uy becomes G times T, and Uy becomes G times T as well, being they are both equal for all values of T. This means they're always for the same velocity as each other. We can also look at a position. The distance of bullet travels vertically, which is V0Y times T, plus half of GT squared, we're now equal to half of GT squared. This will also be the same for the duck. Again, we didn't always travel the same vertical distance in the same time, so I always add the same height. So I always add the same height. When the duck and the bullet is applied, they will both be at D. This means that the time taken for the bullet to reach the same horizontal position as the duck is D over V0X. This is the time at which the bullet clies with the duck. Mario triumphantly sees the duck falling. However, he observes that it stops accelerating at some point before hitting the ground. He wonders why. Once it starts falling on the gravity and air friction, once it starts falling on the gravity and air friction act on the duck. The force due to gravity is M times G. The force due to air friction is minus beta times V, where beta is the friction coefficient and V is the velocity of the duck. We take downward-dispositive y-direction, and that is y-sinus here as friction acts upward, reducing the total acceleration. The total force fell by the duck is the sum of these two forces. This is equal to the mass of the duck times its total acceleration, and here we have our equation of motion. Acceleration is the first derivative of velocity with respect of time, as the velocity is dependent on time, and so we substitute that into our equation of motion. We can rearrange our expression by multiplying both sides by DT or M, and also dividing both sides by the expression we have on the left hand side before that. After that, we can factor out the beta in the denominator, so we have V with respect to T alone. We multiply both sides by minus beta, and so we have an expression on the right only dependent on DV and on the left hand side we have an expression only dependent on DT. Mario triumphacy sees the duck falling. However, he observes that it stops accelerating at some point before hitting the ground. He wonders why. Once it starts falling, only gravity and air friction act on the duck. The force due to gravity's M is... We integrate both sides so that we have an expression dependent on time and velocity, and we obtain this expression here. We exponentiate it so that E and the natural log cancel out, and we are ranging, we obtain an expression for the velocity. We have this arbitrary constant A that comes from the constant we add when integrating. This constant can be determined by the initial conditions. In this general equation, we can find that A equals this expression here, and our final solution for the velocity of the duck if it starts with zero velocity in the y-axis is this. We can see that as time becomes larger and larger, this term goes to zero. Thus the velocity stops increasing and reaches its specific value, the largest it can be. This is called terminal velocity, and at this point the two forces, air friction and gravity are in equilibrium, and so the duck stops accelerating. The second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck. The second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck, and the second force is the velocity of the duck, and the