Question

A particle moves with position r(t) = x(t) i + y(t) j where x(t) = 10t² and y(t) = -3t + 2, with x and y in meters and t in seconds.

(a) Find the average velocity for the time interval from 1.00 s to 3.00 s.
(b) Find the instantaneous velocity at t = 1.00 s.
(c) Find the average acceleration from 1.00 s to 3.00 s.
(d) Find the instantaneous acceleration at t = 1.00 s.

Answer 1

given :

a) at t=1s,

xi= 10 m and yi = -1 m

at t=3s,

xf = 90 m and yf = -7 m

average velocity along x = (xf - xi) / (3-1) = (90-10)/(3-1) = 40 m/s

average velocity along y = (yf - yi) / (3-1) = (-7 - (-1)) / (3-1) = -3 m/s

therefore, Net average velocity = [answer]

b) derivative of x with respect to time gives instantaneous velocity along x .

at t=1s, vx = 20 m/s

derivative of y with respect to time gives instantaneous velocity along y .

at t=1s, vy = -3 m/s

Therefore, net instantaneous velocity at t = 1 s, is [answer]

c) at t = 1 s, and

at t = 3s, and

Average acceleration along x =

Average acceleration along y =

Therefore, Net average acceleration = [answer]

d) the derivative of with time gives instantaneous acceleration along x =

at t = 1,

the derivative of with time gives instantaneous acceleration along y =

at t = 1,

Therefore, net instantaneous acceleration = [answer]