Motion In Two Dimensions

Kinematics of Class 11

Motion In Two Dimensions

Whatever we have studied in the kinematics of one dimensional motion the same is applicable for motion in two and three dimensions.

In two dimensions the position vector r of a particle whose coordinate are (x, y) is

r = xMotion In Two Dimensions + yMotion In Two Dimensions

If the particle moves from P1 at position r1 to P2 at position r2, as shown in figure its displacement is given by

Δr = r2 − r1 = ΔxMotion In Two Dimensions + ΔyMotion In Two Dimensions

Motion In Two Dimensions

The student must carefully note the correct direction of Δr.

Δr is the vector that must be added to the initial position r1 to give the final position r2,

i.e. r2 = r1 + Δr

The average velocity is defined as the ratio of the displacement over the time interval

vav = Δr/Δt

Motion In Two Dimensions

or v = dr/dt  = vxMotion In Two Dimensions + vyMotion In Two Dimensions(7.13)

where vx = dx/dt and vy = dy/dt

The direction of v is along the tangent to the path.

The instantaneous acceleration is the ratio of change of the velocity with respect to time.

a = Motion In Two Dimensions(7.14)

where a x = Motion In Two Dimensions

Note that one cannot determine acceleration directly from the path of the particle. One needs to know how each component of the velocity varies as function of space and time.

Motion In Two Dimensions

General equations of kinematics for constant acceleration

v = v0 + at(7.15)

r = r0 + v0t + 1/2 at2(7.16)

r = r0 + 1/2 (v0 + v)t(7.17)

For two−dimensional motion in the plane, the x and y components of these equations are:

vx = v0x + axtvy = v0y + ayt

x = x0 + v0x + 1/2 axt2y = y0 + y0yt + 1/2 ayt2

x = x0 + 1/2 (v0x + vx)ty = y0 + 1/2(v0y + vy)t

Motion In Two Dimensions

 

Example : 7.11

A particle moves one quarter of a circular path of radius 20 m in 10 s. Its initial position is given by ri = (20 m)i and its final position is rf = (20m)j .

(a)Find its displacement Δr and average velocity vav

(b)Find the magnitude of its average velocity and its average speed.

Solution

(a)The displacement is given by

Motion In Two Dimensions

orMotion In Two Dimensions

and the average velocity is

Motion In Two Dimensions

Motion In Two Dimensions (m/s)

(b)The magnitude of the average velocity is

Motion In Two Dimensions m/s

Average speed = Motion In Two Dimensions

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