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A particle starts from rest with constant acceleration. The ratio of space-average velocity to the time average velocity is: a) ½ b) ¾ c) 4/3 d) 3/2

A particle starts from rest with constant acceleration. The ratio of space-average velocity to the time average velocity is:
 
a) ½
b) ¾
c) 4/3
d) 3/2
 

Grade:12th pass

2 Answers

Eshan
askIITians Faculty 2095 Points
4 years ago
Dear student,

Space average velocity=\dfrac{\int_0^x vdx}{x}=\dfrac{2a}{3}t
Time average velocity=\dfrac{\int_0^t vdx}{x}=\dfrac{a}{2}t

Hence the required ratio=\dfrac{4}{3}
ankit singh
askIITians Faculty 614 Points
2 years ago

 

Space average velocity is ∫dx∫vdx​...........(a)
Time – average velocity is ∫dt∫vdt​...........(b)
Initially the particle is at rest so u = 0.
v=at.........(1)
x=21​at2........(2)
dx=atdt..............(3)
For space-average velocity put (1) and (3) in
∫atdt∫a2t2dt​=2t2​3at3​​=32​at.........(4)
For time average velocity ∫dt∫atdt​=2tat2​=2at​...........(5)
Taking ratios of (4) and (5)
2at​32​at​=34​
The ratio of space- average velocity to the time average velocity is 4:3

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