Vectors and Calculus.

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Presentation transcript:

Vectors and Calculus

Position, Velocity, Acceleration The position of a particle on a number line is given by the equation p(t) = 𝑡 2 −5𝑡+6 for 𝑡≥0. Find a velocity equation for the particle. Find an acceleration equation for the particle. What is the speed of the particle at time 𝑡=2? Is the particle speeding up or slowing down at this point? At what time does the particle stop?

Position, Velocity, Acceleration A particle moves in a PLANE according to the parametric curve 𝑥 𝑡 =2𝑡−1, 𝑦 𝑡 =2 𝑡 2 −5𝑡+1, where t is time in seconds. Find a position vector for the particle. Find a velocity vector for the particle. Find an acceleration vector for the particle. At what time, if any, does the particle stop? What is the speed of the particle at time 𝑡=5? How far has the particle traveled after 5 seconds?

Acceleration, Velocity, Position An object is dropped from a building (initial velocity is 0) that is 10 meters high (initial position is 10). Write an acceleration equation (Remember that acceleration due to gravity is approximately -9.8 m/sec2. Write a velocity equation. Write a position equation. At what time does the object hit the ground?

Acceleration, Velocity, Position The acceleration vector for a particle moving in a PLANE is 3𝑡, cos 𝑡 , where t is time in seconds. At 𝑡=0, the particle has velocity 5, 1 and is at position (1, −1). Find the velocity vector. Find the position vector. What is the speed of the particle at time 𝑡=3? Is the particle ever at the origin? If so, at what time does this occur?

A.P. BC Test 2001 - Calculator For 0≤𝑡≤3, and object moving along a curve in the xy-plane has position 𝑥 𝑡 , 𝑦(𝑡) with 𝑑𝑥 𝑑𝑡 = sin 𝑡 3 and 𝑑𝑦 𝑑𝑡 =3 cos 𝑡 2 . At time 𝑡=2, the object is at position (4, 5). Write an equation for the line tangent to the curve at (4, 5). Find the speed of the object at time 𝑡=2. Find the total distance traveled by the object over the time interval 0≤𝑡≤1. Find the position of the object at time 𝑡=3.