Key Points

The scalar product of two vectors $\mathbf{A}$ and $\mathbf{B}$ is a scalar quantity defined as $\mathbf{A} \cdot \mathbf{B} = AB \cos \theta$, where $\theta$ is the angle between them. In component form, $\mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z$.

In physics, work is done when a force causes displacement. For a constant force $\mathbf{F}$ causing displacement $\mathbf{d}$, work is defined as the dot product $W = \mathbf{F} \cdot \mathbf{d} = Fd \cos \theta$. Its SI unit is the joule (J).

Work done is positive if the angle $\theta$ between force and displacement is acute ($0^{\circ} \le \theta < 90^{\circ}$), negative if obtuse ($90^{\circ} < \theta \le 180^{\circ}$), and zero if they are perpendicular ($\theta = 90^{\circ}$).

For a force that varies with position, the work done is calculated by integrating the force over the displacement path. In one dimension, $W = \int_{x_i}^{x_f} F(x) \,dx$, which is the area under the force-displacement graph.

Kinetic energy (K) is the energy an object possesses due to its motion. It is a scalar quantity calculated as $K = \frac{1}{2} m v^2$, where $m$ is mass and $v$ is speed. Its SI unit is the joule (J).

This fundamental theorem states that the work done by the net force on an object equals the change in its kinetic energy. The formula is $W_{\text{net}} = K_f - K_i = \Delta K$.

Potential energy (V) is the stored energy an object has due to its position or configuration. It is defined only for conservative forces.

A force is conservative if the work done by it is independent of the path taken and depends only on the initial and final positions. Examples include gravitational force and spring force.

For a conservative force, the force can be derived from the potential energy function. In one dimension, the relationship is $F(x) = -\frac{dV(x)}{dx}$.

Gravitational potential energy near the Earth's surface is $V(h) = mgh$. The elastic potential energy stored in a spring is $V(x) = \frac{1}{2} k x^2$, where $k$ is the spring constant and $x$ is the displacement from equilibrium.

If only conservative forces are doing work on a system, the total mechanical energy (sum of kinetic and potential energy) remains constant. This is expressed as $K_i + V_i = K_f + V_f$.

Power (P) is the rate at which work is done or energy is transferred. Average power is $P_{\text{av}} = \frac{W}{t}$, and instantaneous power is $P = \frac{dW}{dt} = \mathbf{F} \cdot \mathbf{v}$. Its SI unit is the watt (W).

In any collision, isolated from external forces, the total linear momentum of the system is always conserved. This means the total momentum before the collision equals the total momentum after.

In an elastic collision, both total momentum and total kinetic energy are conserved. In an inelastic collision, total momentum is conserved, but some kinetic energy is lost, usually as heat or sound.

A completely inelastic collision is one where the objects stick together after impact and move with a common final velocity. This type of collision involves the maximum possible loss of kinetic energy.