Kinematics describes motion in terms of position, velocity, and acceleration.

General kinematic relationships are given mathematically by:

Instantaneous velocity $v_s=ds\text{/}dt=$ slope of position graph

Instantaneous acceleration $a_s=d{v}_s\text{/}dt=$ slope of velocity graph

Final position $s_{\text{f}}=s_{\text{i}}+{\displaystyle {\int }_{t_{\text{i}}}^{t_{\text{f}}}{v}_{s}dt=s_{\text{i}}+\{\text{area under the velocity curve from }{t}_{\text{i}}\text{to }{t}_{\text{f}}}$

Final velocity $v_{\text{f}s}=v_{\text{i}s}+{\displaystyle {\int }_{t_{\text{i}}}^{t_{\text{f}}}{a}_{s}dt=v_{\text{i}s}+}\{\text{area under the acceleration curve from }{t}_{\text{i}}\text{to }{t}_{\text{f}}$

Position, velocity, and acceleration are related graphically.

• The slope of the position-versus-time graph is the value on the velocity graph.

• The slope of the velocity graph is the value on the acceleration graph.

• s is a maximum or minimum at a turning point, and $v_s=0$.

  

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• Displacement is the area under the velocity curve.

  

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The sign of $v_s$ indicates the direction of motion.

• $v_s>0$ is motion to the right or up.

• $v_s<0$ is motion to the left or down.

The sign of $a_s$ indicates which way $\vec{a}$ points, not whether the object is speeding up or slowing down.

• $a_s>0$ if $\vec{a}$ points to the right or up.

• $a_s<0$ if $\vec{a}$ points to the left or down.

• The direction of $\vec{a}$ is found with a motion diagram.

An object is speeding up if and only if $v_s$ and $a_s$ have the same sign.

An object is slowing down if and only if $v_s$ and $a_s$ have opposite signs.

Free fall is constant-acceleration motion with

Motion on an inclined plane has $a_s=\pm g\text{sin}\theta$. The sign depends on the direction of the tilt.