Key Points

Motion is the change in an object's position over time. An object can be considered a point object if its size is negligible compared to the distance it travels.

Path length is the total distance covered and is a scalar quantity. Displacement is the shortest distance between the initial and final points, $\Delta x = x_f - x_i$, and is a vector quantity.

Average velocity is the displacement divided by the time interval, $\bar{v} = \frac{\Delta x}{\Delta t}$. Average speed is the total path length divided by the time interval. Average speed is always greater than or equal to the magnitude of average velocity.

Instantaneous velocity is the velocity of an object at a specific instant. It is the first derivative of position with respect to time, $v = \frac{dx}{dt}$.

Instantaneous speed is the magnitude of the instantaneous velocity at a particular instant. It is always equal to the magnitude of instantaneous velocity.

Average acceleration is the change in velocity over a time interval, $\bar{a} = \frac{\Delta v}{\Delta t}$. Instantaneous acceleration is the rate of change of velocity at an instant, $a = \frac{dv}{dt}$.

For a position-time ($x-t$) graph, the slope at any point gives the instantaneous velocity. A straight line indicates constant velocity, while a curved line indicates acceleration.

For a velocity-time ($v-t$) graph, the slope represents acceleration, and the area under the curve represents displacement over a given time interval.

This equation relates final velocity ($v$), initial velocity ($v_0$), acceleration ($a$), and time ($t$). The formula is $v = v_0 + at$.

This equation relates displacement ($x$), initial velocity ($v_0$), time ($t$), and acceleration ($a$). The formula is $x = v_0 t + \frac{1}{2}at^2$.

This equation relates final velocity ($v$), initial velocity ($v_0$), acceleration ($a$), and displacement ($x$). The formula is $v^2 = v_0^2 + 2ax$.

An object in free fall experiences a constant downward acceleration due to gravity, $g \approx 9.8 \text{ m/s}^2$. The kinematic equations are applicable with $a = -g$ (taking the upward direction as positive).

An object can have zero velocity at an instant and still have non-zero acceleration. For example, a ball thrown upwards has zero velocity at its highest point, but its acceleration is still $-g$.

The stopping distance ($d_s$) is the distance a vehicle travels before coming to rest. It is given by $d_s = \frac{-v_0^2}{2a}$, showing it is proportional to the square of the initial velocity.

Reaction time is the time a person takes to observe, think, and act. For a falling object caught after distance $d$, the reaction time is $t_r = \sqrt{\frac{2d}{g}}$.