Chapter 4: Acceleration

Acceleration Change in velocity in a unit of time interval

            a = v/t = (v₂ - v₁)/(t₂ - t₁)

Acceleration = Slope of velocity-time graph

Constant Acceleration

            a = v/t = (v₂ - v₁)/(t₂ - t₁)

            v₂ - v₁ = a(t₂ - t₁)

            v₂ = v₁ + a(t₂ - t₁)

If t₁ = 0, then:

            v₂ = v₁ + at        -------   Use to determine velocity as a function of time. 

Recall:

            v_av = (d₂ - d₁)/(t₂ - t₁)

            d₂ - d₁ = v_av(t₂ - t₁)

If t₁ = 0, then:

            d₂ - d₁ = v_avt

            d₂ = d₁ +  v_avt

            v_av = ½(v₁ + v₂)

            d₂ = d₁ +  ½(v₁ + v₂)t   ------ Use to determine position as a function of time.

            ½(v₂ + v₁) = ½(v₁ + (v₁ + at)) = v₁ + ½at

            d₂ = d₁ +  ½(v₁ + v₂)t =  d₁ + (v₁ + ½at)t

            d₂ = d₁ + v₁t + ½at²     -------Use to determine position as a function of time.

Recall:

            v₂ = v₁ + at      or     t = (v₂ - v₁)/a     

            and

            d₂ = d₁ +  ½(v₁ + v₂)t  

Therefore:

            d₂ = d₁ +  ½(v₁ + v₂)(v₂ - v₁)/a        

            d₂ = d₁ +  (v₂² - v₁²)/2a  

Therefore:

            v₂²  = v₁²  + 2a(d₂ - d₁)   ------- Use to determine velocity as a function of position

Summary of Constant-Acceleration Motion Equations

                       v₂ = v₁ + at    

                       v_av = ½(v₁ + v₂)  

                       d₂ = d₁ +  ½(v₁ + v₂)t  

                       d₂ = d₁ + v₁t + ½at²     

                       v₂²  = v₁²  + 2a(d₂ - d₁)  

Freely Falling Objects

                       a = g = 9.81 m/s²