NSW HSC Physics

## Two independent motions A projectile is an object moving under the influence of gravity alone after it has been launched. Its path is curved, but the curve is easy to analyse because the motion splits into a horizontal part and a vertical part that do not affect each other. The force of gravity acts straight down, so it changes only the vertical velocity. Nothing pushes sideways, so the horizontal velocity stays as it was. The analysis rests on two assumptions. - The vertical acceleration is constant and equal to the acceleration due to gravity, $g = 9.8\,\mathrm{m\,s^{-2}}$ downward. This is reasonable when the height gained is small compared with the radius of the Earth. - There is no air resistance. In reality drag shortens the range and lowers the maximum height, especially for light objects and high speeds, so the predictions are upper limits for a real projectile. Choosing the launch point as the origin, with up as positive, a projectile launched with speed $u$ at an angle $\theta$ above the horizontal has initial velocity components $$ u_x = u\cos\theta, \qquad u_y = u\sin\theta $$ $u_x$ and $u_y$ are the horizontal and vertical components in metres per second, and $\theta$ is measured from the horizontal. ## Equations for each direction The horizontal motion is at constant velocity, and the vertical motion is at constant acceleration $a_y = -g$. | Horizontal | Vertical | |---|---| | $v_x = u_x$ (constant) | $v_y = u_y - gt$ | | $x = u_x t$ | $y = u_y t…