Translational Motion
Definition
Translational motion refers to motion in which all points of a rigid body move with the same velocity.
Explanation
In the above definition one could use the term “body” instead of “rigid body”, but an object that maintains translational motion is effectively equivalent to a rigid body. Therefore, an object undergoing translational motion is assumed to be a rigid body.
Put another way, it means that the velocity vectors at all points within the object are identical. Let the velocity at an arbitrary point in the object be $\mathbf{v}_{i}$,
$$ \mathbf{v}_{i}(t) = \mathbf{v}_{j}(t) \quad \forall i \ne j $$
Hence the object does not rotate, and the acceleration of every point is also identical. The trajectory of motion may be a straight line or a curve. The momentum of a translating object is called linear momentum.
Characteristics
The velocity and acceleration of every point in the object are identical. $$ \mathbf{v}_{i} = \mathbf{v}_{j}, \quad \mathbf{a}_{i} = \mathbf{a}_{j} $$
The shape and orientation of the object do not change. The angular velocity is $\mathbf{0}$. $$ \omega = \mathbf{0} $$ That is, there is no rotation; if one considers the line segment joining any two points in the object, the direction of that segment does not change with time.
The motion can be simply described by the motion of the center of mass. $$ \mathbf{p} = m \mathbf{v}_{\mathrm{cm}} $$ $$ \mathbf{F} = m \mathbf{a}_{\mathrm{cm}} $$ $$ T = \dfrac{1}{2}m v_{\mathrm{cm}}^2 $$