Projectile Motion Review

Two-Dimensional Motion: Core Idea

• In 2D motion, analyze horizontal (x) and vertical (y) components separately, treating each as its own one-dimensional problem.

• Time t is common to both directions; what happens in x happens over the same time interval as what happens in y.

• For ideal cases (negligible air resistance), the two directions do not influence each other directly; you can apply 1D kinematics to each axis independently and then combine results as vectors.

• Displacement, velocity, and acceleration can be expressed for each axis: x-components and y-components carry their own values and signs.

Coordinate System and Variables

• Horizontal displacement: $x(t)$; initial position: $x<em>0$; initial velocity component: $v</em>{0x}$; acceleration component: $a_x$.

• Vertical displacement: $y(t)$; initial position: $y<em>0$; initial velocity component: $v</em>{0y}$; acceleration component: $a_y$.

• Time: $t$; note that one second in x is the same as one second in y (time is scalar, not direction-specific).

• Velocities at time t: $v<em>x(t) = v</em>{0x} + a<em>x t$ and $v</em>y(t) = v<em>{0y} + a</em>y t$.

• Displacements along each axis follow: $x = x<em>0 + v</em>{0x} t + frac{1}{2} a<em>x t^2$ and $y = y</em>0 + v<em>{0y} t + frac{1}{2} a</em>y t^2$.

• If both axes are considered together, the overall velocity is a vector oxed{oldsymbol{v} = ig(vx, vyig)} and the speed is its magnitude |oldsymbol{v}| =
  vert ig(vx, vyig)
  angle =
  olinebreak \ \ \
  abla o ext{or simply } |oldsymbol{v}| = \
  olinebreak \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }