12.6: Vector Magnitude & Distance

Warmup If velocity of a particle is given by 𝑣 𝑡 =arcsin⁡(2𝑥), calculate the distance the particle traveled over the interval (0,.5).

A vector is a quantity have both ______________ and ______________. So if we wanted to find the magnitude of both the x(t) and y(t) vectors together. What would we do? (Campbell: for visual help draw a graph with two vectors)

𝑥 𝑡 ,𝑦 𝑡 = (𝑥 𝑡 ) 2 +(𝑦(𝑡) ) 2 Determining vector magnitude 𝑥 𝑡 ,𝑦 𝑡 = (𝑥 𝑡 ) 2 +(𝑦(𝑡) ) 2 Does this vector magnitude give you direction?

𝑥′′ 𝑡 ,𝑦′′ 𝑡 = (𝑥′′ 𝑡 ) 2 +(𝑦′′ 𝑡 ) 2 = So what would each of these vector magnitudes give you? 𝑥 𝑡 ,𝑦 𝑡 = (𝑥 𝑡 ) 2 +(𝑦 𝑡 ) 2 = 𝑥′ 𝑡 ,𝑦′ 𝑡 = (𝑥′ 𝑡 ) 2 +(𝑦′ 𝑡 ) 2 = 𝑥′′ 𝑡 ,𝑦′′ 𝑡 = (𝑥′′ 𝑡 ) 2 +(𝑦′′ 𝑡 ) 2 =

𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒= 𝑎 𝑏 (𝑥′ 𝑡 ) 2 +(𝑦′ 𝑡 ) 2 𝑑𝑥 How would you calculate distance traveled? 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒= 𝑎 𝑏 (𝑥′ 𝑡 ) 2 +(𝑦′ 𝑡 ) 2 𝑑𝑥

A particle moves along a curve defined by the function 𝑦=(𝑥−3 ) 3 2 ∙ 𝑥+2 . The x-coordinate of the particle is given by the function 𝑥(𝑡), which is satisfied by the equation 𝑑𝑥 𝑑𝑡 = 2 𝑡−3 for 𝑡≥0 with the initial condition 𝑥 4 =1. Find the particular solution for 𝑥 𝑡 . (by hand) Find 𝑑𝑦 𝑑𝑡 in terms of t. (by hand)

Find the speed of the particle at t=6 (calculator) Find the total distance traveled by the particle from 4,6 . (calculator) Find the acceleration vector at t=6. (calculator)