12.5: Vector PVA

What is a vector? A vector is a quantity with magnitude and direction. A parametric equation calculates 2 vectors. For example, a baseball is hit and the horizontal position is given by 𝑥=3𝑡 and the vertical position is determined by 𝑦=− 𝑡 2 +3𝑡. A position vector is written (𝑥 𝑡 ,𝑦 𝑡 ) or <𝑥 𝑡 ,𝑦 𝑡 >

How do you think you would find the velocity vector? A velocity vector is written (𝑥′ 𝑡 ,𝑦′ 𝑡 ) or < 𝑥 ′ (𝑡), 𝑦 ′ (𝑡)> 𝑥 ′ 𝑡 = 𝑑𝑥 𝑑𝑡 𝑦 ′ 𝑡 = 𝑑𝑦 𝑑𝑡 How do you find the acceleration vector? A velocity vector is written (𝑥′′ 𝑡 ,𝑦′′ 𝑡 ) or < 𝑥 ′′ (𝑡), 𝑦 ′′ (𝑡)> 𝑥 ′ 𝑡 = 𝑑 2 𝑥 𝑑 𝑡 2 𝑦 ′ 𝑡 = 𝑑 2 𝑦 𝑑 𝑡 2

How would you determine the motion of an object to the left or right? 𝑥′(𝑡) How would you determine the motion of an object up or down? y′(𝑡)

Calculator Position is given by the parametric equations 𝑥= 𝑡 2 −ln⁡(1+𝑡) and 𝑦=2 cos 𝑡 for the interval [0,𝜋]. Find the position of the particle at t= 𝜋 2 Find the velocity vector at t= 𝜋 2 . Find the time at which x(t) has it’s minimum value, and calculate its velocity vector. Find when the particles horizontal motion is to the left.

Calculator A particle moves along a curve such that that the position is given by 𝑥 𝑡 ,𝑦 𝑡 . The particle’s velocity vecor is given by 𝑣 𝑡 =( sin 2𝑡 , 𝑒 −𝑡 ). At t=0, the particles position is (2,4). Find the particles position at t=1 Find the acceleration vector at t=2 At what time does the tangent line to the particles path of motion have a slope of 4?