Normal acceleration of B:
( a B ) n = v B 2 p = 2 0 2 0.3 = 1333.34 k m / h 2 N o r m a l a c c e l e r a t i o n v e c t o r o f B : ( a B ) n = ( a b ) n [ c o s 30 ° i − s i n 30 ° j ] = 1333.34 [ c o s 30 ° i − s i n 30 ° j ] = 1154.67 i − 666.67 j (a_B)_n = \frac{v_B^2}{p} = \frac{20^2}{0.3}=1333.34 \;km/h^2 \\
Normal\; acceleration\; vector \; of \; B: \\
(a_B)_n = (a_b)_n[cos 30°i -sin 30°j] \\
= 1333.34[cos 30°i -sin 30°j] \\
= 1154.67i -666.67j ( a B ) n = p v B 2 = 0.3 2 0 2 = 1333.34 km / h 2 N or ma l a cce l er a t i o n v ec t or o f B : ( a B ) n = ( a b ) n [ cos 30° i − s in 30° j ] = 1333.34 [ cos 30° i − s in 30° j ] = 1154.67 i − 666.67 j
Tangential acceleration vector of B
( a B ) t = ( a B ) t [ − s i n 30 ° i − c o s 30 ° j ] = 800 [ − s i n 30 ° i − c o s 30 ° j ] = − 400 i − 692.8 j T o t a l a c c e l e r a t i o n o f B a B = ( a B ) t + ( a B ) n = − 400 i − 692.8 j + 1154.67 i − 666.67 j = 754.67 i − 1359.47 j A c c e l e r a t i o n o f B w i t h r e s p e c t t o A . a B / A = a B − a A = ( 754.67 i − 1359.47 j ) − 400 j = 754.67 i − 1759.47 j a B / A = ( 754.67 ) 2 + ( 1759.47 ) 2 = 1914.487 k m / h 2 θ = t a n − 1 ( − 1759.47 754.67 ) θ = − 66.784 ° (a_B)_t = (a_B)_t [-sin30°i -cos30°j] \\
= 800[-sin30°i -cos30°j] \\
= -400i -692.8j \\
Total \;acceleration \;of \;B \\
a_B = (a_B)_t +(a_B)_n \\
= -400i -692.8j +1154.67i -666.67j \\
= 754.67i -1359.47j \\
Acceleration \; of\; B\; with \; respect\; to \; A. \\
a_{B/A} = a_B -a_A \\
= (754.67i -1359.47j) -400j \\
= 754.67i -1759.47j \\
a_{B/A} = \sqrt{(754.67)^2 +(1759.47)^2} \\
= 1914.487 \;km/h^2 \\
θ = tan^{-1} (\frac{-1759.47}{754.67}) \\
θ = -66.784° ( a B ) t = ( a B ) t [ − s in 30° i − cos 30° j ] = 800 [ − s in 30° i − cos 30° j ] = − 400 i − 692.8 j T o t a l a cce l er a t i o n o f B a B = ( a B ) t + ( a B ) n = − 400 i − 692.8 j + 1154.67 i − 666.67 j = 754.67 i − 1359.47 j A cce l er a t i o n o f B w i t h res p ec t t o A . a B / A = a B − a A = ( 754.67 i − 1359.47 j ) − 400 j = 754.67 i − 1759.47 j a B / A = ( 754.67 ) 2 + ( 1759.47 ) 2 = 1914.487 km / h 2 θ = t a n − 1 ( 754.67 − 1759.47 ) θ = − 66.784°
Velocity
v B = − v b c o s 30 ° i + v B s i n 30 ° j = − 20 c o s 30 ° i + 20 s i n 30 ° j v B = − 17.32 i + 10 j v A = − v A i = − 30 i v B = v A + v B / A v B / A = v B − v A = − 17.32 i + 10 j + 30 i = 12.68 i + 10 j = ( 12.68 ) 2 + ( 10 ) 2 = 16.148 k m / h θ = t a n − 1 ( 10 12.68 ) θ = 38.26 ° v_B = -v_bcos 30°i+v_Bsin30°j \\
= -20cos30°i +20sin30°j \\
v_B = -17.32i +10j \\
v_A = -v_Ai =-30i \\
v_B =v_A +v_{B/A} \\
v_{B/A} = v_B -v_A \\
= -17.32i +10j +30i \\
= 12.68i +10j \\
= \sqrt{(12.68)^2+(10)^2} \\
= 16.148 \;km/h \\
θ = tan^{-1}(\frac{10}{12.68}) \\
θ = 38.26° v B = − v b cos 30° i + v B s in 30° j = − 20 cos 30° i + 20 s in 30° j v B = − 17.32 i + 10 j v A = − v A i = − 30 i v B = v A + v B / A v B / A = v B − v A = − 17.32 i + 10 j + 30 i = 12.68 i + 10 j = ( 12.68 ) 2 + ( 10 ) 2 = 16.148 km / h θ = t a n − 1 ( 12.68 10 ) θ = 38.26°
#332078 on Oct 2022
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