University Physics Volume 1

# Key Equations

### Key Equations

 Position vector $r→(t)=x(t)i^+y(t)j^+z(t)k^r→(t)=x(t)i^+y(t)j^+z(t)k^$ Displacement vector $Δr→=r→(t2)−r→(t1)Δr→=r→(t2)−r→(t1)$ Velocity vector $v→(t)=limΔt→0r→(t+Δt)−r→(t)Δt=dr→dtv→(t)=limΔt→0r→(t+Δt)−r→(t)Δt=dr→dt$ Velocity in terms of components $v→(t)=vx(t)i^+vy(t)j^+vz(t)k^v→(t)=vx(t)i^+vy(t)j^+vz(t)k^$ Velocity components $vx(t)=dx(t)dtvy(t)=dy(t)dtvz(t)=dz(t)dtvx(t)=dx(t)dtvy(t)=dy(t)dtvz(t)=dz(t)dt$ Average velocity $v→avg=r→(t2)−r→(t1)t2−t1v→avg=r→(t2)−r→(t1)t2−t1$ Instantaneous acceleration $a→(t)=limt→0v→(t+Δt)−v→(t)Δt=dv→(t)dta→(t)=limt→0v→(t+Δt)−v→(t)Δt=dv→(t)dt$ Instantaneous acceleration, component form $a→(t)=dvx(t)dti^+dvy(t)dtj^+dvz(t)dtk^a→(t)=dvx(t)dti^+dvy(t)dtj^+dvz(t)dtk^$ Instantaneous acceleration as secondderivatives of position $a→(t)=d2x(t)dt2i^+d2y(t)dt2j^+d2z(t)dt2k^a→(t)=d2x(t)dt2i^+d2y(t)dt2j^+d2z(t)dt2k^$ Time of flight $Ttof=2(v0sinθ0)gTtof=2(v0sinθ0)g$ Trajectory $y=(tanθ0)x−[g2(v0cosθ0)2]x2y=(tanθ0)x−[g2(v0cosθ0)2]x2$ Range $R=v02sin2θ0gR=v02sin2θ0g$ Centripetal acceleration $aC=v2raC=v2r$ Position vector, uniform circular motion $r→(t)=Acosωti^+Asinωtj^r→(t)=Acosωti^+Asinωtj^$ Velocity vector, uniform circular motion $v→(t)=dr→(t)dt=−Aωsinωti^+Aωcosωtj^v→(t)=dr→(t)dt=−Aωsinωti^+Aωcosωtj^$ Acceleration vector, uniform circular motion $a→(t)=dv→(t)dt=−Aω2cosωti^−Aω2sinωtj^a→(t)=dv→(t)dt=−Aω2cosωti^−Aω2sinωtj^$ Tangential acceleration $aT=d|v→|dtaT=d|v→|dt$ Total acceleration $a→=a→C+a→Ta→=a→C+a→T$ Position vector in frameS is the positionvector in frame $S′S′$ plus the vector from theorigin of S to the origin of $S′S′$ $r→PS=r→PS′+r→S′Sr→PS=r→PS′+r→S′S$ Relative velocity equation connecting tworeference frames $v→PS=v→PS′+v→S′Sv→PS=v→PS′+v→S′S$ Relative velocity equation connecting morethan two reference frames $v→PC=v→PA+v→AB+v→BCv→PC=v→PA+v→AB+v→BC$ Relative acceleration equation $a→PS=a→PS′+a→S′Sa→PS=a→PS′+a→S′S$