Inertial motion in an accelerating frame (${\mathbf{a}}^{\prime }=\mathbf{a}-\ddot{\mathbf{R}}$)

Constant velocity depends on the reference frame. For nonrotating axes and a moving origin of laboratory position $\mathbf{R}(t)$, subtracting that origin givesIf $\mathbf{R}(t)={\mathbf{R}}_0+\mathbf{U}t$ with constant velocity $\mathbf{U}$, the Galilean transformation leaves acceleration unchanged. The cart moving at $1.5m/s$ in the laboratory is at rest in a frame moving right at $1.5m/s$. Both descriptions obey Newton's first law.

Instead choose an origin with $R_x(t)=\frac{1}{2}A{t}^2$, where the stipulated constant $A=0.50m/{\mathrm{s}}^2$ points right. The same cart hasAt $t=3\mathrm{s}$, the cart is instantaneously at rest relative to this accelerating origin, but its relative acceleration is nonzero. In this translating accelerated reference frame, Newton's second law must include the effective inertial force $-mA=-1.0\mathrm{N}$. This term accounts for the accelerating coordinates; the actual external forces on the cart still sum to zero. No rotation-dependent terms are present because the axes remain parallel to the laboratory axes.