what is the direction of the object's velocity as it moves along a circular path
Circular motion
a = v  ̇ = i ω  ̇ z + i ω z  ̇ = ( i ω  ̇ − ω 2 ) z = ( i ω  ̇ − ω 2 ) R e i θ ( t ) = − ω 2 R e i θ ( t ) + ω  ̇ e i π 2 R e i θ ( t ) . ( \ displaystyle ( \ begin ( aligned ) a& = ( \ dot ( v ) ) = i ( \ dot ( \ omega ) ) z + i \ omega ( \ dot ( z ) ) = \ left ( i ( \ dot ( \ omega ) ) - \ omega ^ ( 2 ) \ right ) z \ \ & = \ left ( i ( \ dot ( \ omega ) ) - \ omega ^ ( 2 ) \ right ) Re ^ ( i \ theta ( t ) ) \ \ & = - \ omega ^ ( 2 ) Re ^ ( i \ theta ( t ) ) + ( \ dot ( \ omega ) ) e ^ ( i ( \ frac ( \ pi ) ( 2 ) ) ) Re ^ ( i \ theta ( t ) ) \ . \ end ( aligned ) ) )
a = v  ̇ = i ω  ̇ z + i ω z  ̇ = ( i ω  ̇ − ω 2 ) z = ( i ω  ̇ − ω 2 ) R e i θ ( t ) = − ω 2 R e i θ ( t ) + ω  ̇ e i π 2 R e i θ ( t ) . ( \ displaystyle ( \ begin ( aligned ) a& = ( \ dot ( v ) ) = i ( \ dot ( \ omega ) ) z + i \ omega ( \ dot ( z ) ) = \ left ( i ( \ dot ( \ omega ) ) - \ omega ^ ( 2 ) \ right ) z \ \ & = \ left ( i ( \ dot ( \ omega ) ) - \ omega ^ ( 2 ) \ right ) Re ^ ( i \ theta ( t ) ) \ \ & = - \ omega ^ ( 2 ) Re ^ ( i \ theta ( t ) ) + ( \ dot ( \ omega ) ) e ^ ( i ( \ frac ( \ pi ) ( 2 ) ) ) Re ^ ( i \ theta ( t ) ) \ . \ end ( aligned ) ) )