2.7.1 Relative Velocity of A w.r.t. B
Let (X, Y) is a fixed coordinate system and two particles A and B are having position vectors r;: and r;. If we want to find out the relative velocity of A with respect to B, we introduce (x, y), a translating axes. Keeping B point as its moving origin.
From Figure 2.8, we have
where the position vector of A with respect to B.
Differentiating (2.24) with respect to time, we get
Example 2.1 At time t = 0, the position vector of a particle moving in the (x,y) plane is r = 10 I m by time t = 0.04s ,its position vector has become 10 2i + 0.8 j m .find out the magnitude v av the average velocity during this interval of time and angle θ made by the average velocity with the positive x axis.
∆t = (0.04) -0) s = 0.04 s
tan θ = 20 /5 =4 . θ = tan-1 (4) = 75.96o
Example 2.2. A particle which moves in 2-D motion has coordinates given in metres by x = 2t2 -Bt + 40 and y = 6 sin 2t, where t is in seconds. Find out the magnitudes of the velocity V and the acceleration a and the angle B between V and at time t = 3s.
Solution. x = 2t2 – 8t +40
vx =dx /dt = 4t – 8
vx at t = 3 S ,= 4 x 3 – 8 =4 m/s
vy = dy /dt =12 cos 2t
at t =3 ,vy = 12 cos 2×3
= 12 cos 2x 3 x180o = 11.522 m/s
ax = dvx / dt = 4 m /s2
From (4) => ay = dv y /dt = -24 sin 2t
= -24 sin 2 x3 x 180o / π
= 6.706 m/ s2