Position, Velocity, Displacement, and Distance Definition: Position, Velocity, Displacement, and Distance: a. The position of an object moving along a line at time t, denoted sit), is the location of the object relative to the origin. b. The velocity of an object at time t is vit)-s'(t). c. The displacement of the object between t-a and t-b>a is s(b)-s(a)= -fv(1) dt. d. The distance traveled by the object between tea and t-ba is vildt, where (1) is the speed of the object at time t. 6+ ++ 0 HD-2-M+6 Displacement (1) dr (1) c Ama= Displacement from/-0 tor-3-1-0 Example 1: Given the velocity function v(t)=1-61+8 on 0ts5, where t is measured in seconds and v(t) is measured in m/s. a. Graph the function over the given interval. Determine when the motion is in the positive direction and when it is in the negative direction. Displacement -Land - b. Find the displacement over the given interval. c. Find the distance traveled over the given interval. 64 -12-6 is Area- Ares- Distance traveled from 70 tor3 Position from Velocity: Given the velocity v(t) of an object moving along a line and its initial position s(0), the position function of the object for future timest 20 is jv(x)dx Position at t = v(t) = velocity at t s(0) + Initial position Velocity from Acceleration: Given the velocity a(t) of an object moving along a line and its initial velocity v(0), the velocity function of the object for future times t O is 0 displacement over [0] v(0) + ja(x)dx initial velocity 0 velocity over [0,1] Example 2: Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. a(t)=-32, v(0)=50, s (0)=15 Example 3: The acceleration function (in m/s^2) and the initial velocity are given for a particle moving along a line. Find (a) the velocity at time t and (b) the distance traveled during the given time: a(t)=21+3, v(0)=-4,05153