Projectile Motion

• For any constant acceleration: \(d = v_0t + \frac{1}{2}at^2\)
• For any special case of horizontal projection:

\begin{equation*} \begin{split} a_x = 0; \\ a_y = g; \\ v_{0y} = 0; \\ v_{0x} = v_0 \end{split} \end{equation*}

• Horizontal displacement: \(d_x = v_{0x}t\)
• Vertical displacement: \(d_y = \frac{1}{2}gt^2\)

• For any constant acceleration: \(v_f = v_0 + at\)
• For any special case of horizontal projection:

\begin{equation*} \begin{split} a_x = 0; \\ a_y = g; \\ v_{0y} = 0; \\ v_{0x} = v_0 \end{split} \end{equation*}

• Horizontal displacement: \(v_x = v_{0x}\)
• Vertical displacement: \(v_y = gt\)

• The components of displacement at time \(t\) are:

\begin{equation*} \begin{split} d_x = v_{0x}t + \frac{1}{2}a_xt^2 \\ d_y = v_{0y}t + \frac{1}{2}a_yt^2 \end{split} \end{equation*}

• For projectiles:

\begin{equation*} \begin{split} a_x & = 0; \\ a_y & = g; \\ v_{0y} & = v_0 sin\theta; \\ v_{0x} & = v_0 cos\theta \end{split} \end{equation*}

• displacement components x and y for projectiles are:

\begin{equation*} \begin{split} d_x & = v_{0x}t \\ d_y & = v_{0y}t + \frac{1}{2}gt^2 \end{split} \end{equation*}

• The components of velocity at time \(t\) are:

\begin{equation*} \begin{split} v_x = v_{0x} + a_xt \\ v_y = v_{0x} + a_yt \end{split} \end{equation*}

• For projectiles:

\begin{equation*} \begin{split} a_x & = 0; \\ a_y & = g; \\ v_{0y} & = v_0 sin\theta; \\ v_{0x} & = v_0 cos\theta \end{split} \end{equation*}

• displacement components x and y for projectiles are:

\begin{equation*} \begin{split} v_x & = v_{0x}\ constant \\ v_y & = v_{0y} + gt \end{split} \end{equation*}

• It is the force that keeps the object in circular motion.

\begin{equation*} \begin{split} a_c & = \frac{v^2}{R}; \\ F_c & = ma_c = \frac{mv^2}{R} \end{split} \end{equation*}

• where \(m\) is mass of the object, \(v\) is velocity, and \(R\) is radius of the circular path.
• acceleration directed toward the center of circular path.

• when an object moves in a circular path at a distance \(r\) from the center, then the body's velocity is directed tangentially at any instant.
• the linear velocity is its tangential velocity at any instant.

• The rate of change of the tangential velocity of a particular in a circular orbit.

Formula (tho its not in the LAS): \(a_t = ar\)

• \(a_t\) is the tangential acceleration, \(r\) is the radius of the circular path, and \(a\) is the angular acceleration.
• always directs towards the tangent to the path of the body.