Projectile launch: final velocity components, flight time, range, maximum height, impact speed.

00:00 2-D kinematics projectile motion-- projectile launch: final velocity components, flight time, range, maximum height impact speed. In this problem we start with a cannon launching a projectile with initial velocity and height given. The projectile travels into the sea on a parabolic trajectory.

00:46 We prep the problem by clearly setting the origin of coordinates and splitting the initial velocity vector into components. We use the cosine to get the x component of initial velocity, and this velocity is constant throughout the entire flight. We use the sine to get the y component of the initial velocity. Finally, we set the origin at sea level directly below the projectile launch position.

01:32 In the first part of the projectile motion problem, flight time is computed using the vertical equation of motion for the projectile. The reason we start with a vertical analysis is that we have extra information about the vertical position: the height at the moment the projectile lands is zero! We use the constant acceleration kinematics formula y=y_0+v_0*y-1/2*g*t^2 to set up a quadratic equation for time and solve for the flight time. We also discuss the extraneous solution: why is there a negative time solution? The reason is that the mathematical model does not include the physical constraint of the launch moment: the parabolic trajectory starts before the launch, passes through the launch position with exactly the same conditions as the launch, then continues into the future. Finally, we illustrate how to solve the quadratic equation for time by using the computer algebra system wxMaxima.

04:55 Next, we compute the range for the projectile by using the horizontal equation of motion for the projectile. Since we already have the flight time for the projectile and we know the horizontal velocity is constant, we simply use distance=rate*time or x=x_0+v*t to compute the horizontal displacement during the projectile flight. The horizontal displacement is called the "range".

06:06 Next, we compute the maximum height for the projectile by using the fact that the y-velocity is zero at the maximum height. We note that the maximum height does not happen at half the flight time, because this is not a projectile motion on a level surface problem. We apply the kinematics formula v^2=v_0^2+2a(y-y_0), plug in v=0 for the moment of maximum height and solve for the final height.

07:51 Finally, we compute the impact speed of the projectile by using the launch from a cliff final velocity components and the pythagorean theorem. The final x component of velocity is the same as the initial x component of the velocity, and the final y-component of velocity is found by using the kinematics equation v_y=v_0y-gt. Once the final velocity components are determined, we draw a triangle and use the pythagorean theorem to obtain the magnitude of the impact velocity, i.e., the impact speed of the projectile.