A particle possesses two velocities at the same time one 4 m/s due south and other the 2√2 m/s due north- east. Find the magnitude and direction of resultant velocity.

Asked by aditya.loya9821 | 8th Jul, 2021, 09:13: AM

Expert Answer:

Velocity components 4 m/s and 2√2 m/s acting on the object are shown in figure. 
 
Resultant velocity is detrmined by parallelogram method .
 
Let velocity components OA = 4 m/s and OB = 2√2 m/s are two adjacent sides of parallelogram as shown in figure.
 
Let us make the parallelogram OACB with adjacent sides OA and OB as shown in figure.
 
Diagonal OC is resultant velocity that is determined using cosine formula from ΔOBC

OC = OB2 + BC2 - (2 × OB × BC × cos45) 
 
OC2 =  8 + 16 - ( 2 × 4 × 2√2 × cos45 ) = 8
 
Hence , OC = √8 m/s  = 2√2 m/s
 
Since OB = OC , we get begin mathsize 14px style angle end styleOBC = begin mathsize 14px style angle end styleOCB = begin mathsize 14px style angle end styleAOC = 45o
Hence magnitude of resultant = 2√2 m/s and direction is South-East
 
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Resultant OC is determined by vector method as explained below
 
begin mathsize 14px style stack O A with rightwards arrow on top end style =  - 4 begin mathsize 14px style j with hat on top end style m/s  
begin mathsize 14px style stack O B with rightwards arrow on top end style =  ( 2  begin mathsize 14px style i with hat on top end style + 2 begin mathsize 14px style j with hat on top end style )  m/s  
begin mathsize 14px style i with hat on top end style and begin mathsize 14px style j with hat on top end style are unit vectors along x-axis and y-axis direction
Resultant begin mathsize 14px style stack O C with rightwards arrow on top end style = begin mathsize 14px style stack O A with rightwards arrow on top end style + begin mathsize 14px style stack O B with rightwards arrow on top end style  = ( 2  begin mathsize 14px style i with hat on top end style - 2 begin mathsize 14px style j with hat on top end style )  m/s
Magnitude of resultant = | begin mathsize 14px style stack O C with rightwards arrow on top end style | = ( 22 + 22 )1/2 m/s =  2√2 m/s .
Direction of resultant = tan-1 ( uy/ ux ) = tan-1 ( -2/ 2 ) = -45o ,
 
Where uy is y-component of resultant and ux is x-component of resultant
 
i.e. resultant velocity makes angle 45o with x-axis in clockwise direction .
 
Hence direction of resutlat is South-East

Answered by Thiyagarajan K | 8th Jul, 2021, 12:51: PM