Question

Instantaneous velocity vector and accelerstion vector lies on

Answer 1

Concept Map for Velocity and acceleration in two dimensional motion. (THE IMAGE FOR THIS IS ATTACHED BELOW)

Suppose that the position vector function for a particle is given by r(t) = x(t)i + y(t)j  

with x(t) = at + b and y(t) = ct2 + d, where a = 1 m/s, b = 1 m, c = 0.125 m/s2, and d = 1 m.

(a)  Calculate the average velocity during the time interval t = 2 s to t = 4 s.

(b)  Determine the velocity and speed at t = 2 s.

Solution:  

(a)  The position of the particle is given to us as a function of time.  At t = 2 s the position of the particle is

r(2 s) = [(1 m/s)(2 s) + 1 m]i  + [(0.125 m/s2)(4 s2) + 1 m]j = 3 m i + 1.5 m j.

At t = 4 s its position is

r(4 s) = [(1 m/s)(4 s) + 1 m]i + [(0.125 m/s2)(16 s2) + 1 m]j = 5 m i + 3 m j.

The average velocity of the particle between 2 and 4 seconds is  

v = ∆r/∆t = (r(4 s) - r(2 s))/(2 s),

v = [(5 m - 3 m)i + (3 m - 1.5 m)j]/(2 s) = (1 m/s)i + (0.75 m/s)j.

(b)  The instantaneous velocity of the particle is  

v = dr/dt = [d(x(t)/dt]i + [dy(t)/dt]j = ai + 2ctj.