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Simple question about velocity and acceleration

  • Thread starter h_k331
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I'm writing a lab report and came to the following conclusion, I was hoping someone might be able to verify it for me.

The average velocity during the time interval [A,B] will be equal to the instantaneous velocity at the time (A+B)/2 if the acceleration during the time interval [A,B] is a constant.

Thanks,
hk
 

Answers and Replies

  • #2
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yeah, thats true
 
  • #3
Andrew Mason
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h_k331 said:
I'm writing a lab report and came to the following conclusion, I was hoping someone might be able to verify it for me.

The average velocity during the time interval [A,B] will be equal to the instantaneous velocity at the time (A+B)/2 if the acceleration during the time interval [A,B] is a constant.
The graph of velocity - time will have a slope a= v/t that is constant. In other words, it is a straight line with slope a.

The definition of average speed is: [tex]v_{avg} = \frac{distance_{total}}{time_{total}}[/tex]

What is the total distance covered (d = vt) in the time interval [A,B]? What is the total time?

[Hint: think of distance as the area under the graph]

AM
 
  • #4
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daveed said:
yeah, thats true
Thanks Dave.

hk
 
  • #5
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Andrew Mason said:
The graph of velocity - time will have a slope a= v/t that is constant. In other words, it is a straight line with slope a.

The definition of average speed is: [tex]v_{avg} = \frac{distance_{total}}{time_{total}}[/tex]

What is the total distance covered (d = vt) in the time interval [A,B]? What is the total time?

[Hint: think of distance as the area under the graph]

AM
The total distance covered would be the integral of velocity, and the total time would be d/v, right?

hk
 
  • #6
Andrew Mason
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h_k331 said:
The total distance covered would be the integral of velocity, and the total time would be d/v, right?
The time would be B-A.

[tex]t=\frac{distance_{total}}{v_{avg}}= B-A[/tex]

[tex]d = \int_A^B vdt = v_A(B-A) + \frac{1}{2}(v_B - v_A)(B-A)[/tex]

[tex]d/t = v_{avg} = d/(B-A) = v_A + \frac{1}{2}(v_B - v_A) = v_A + \frac{1}{2}(at) [/tex]
[tex]v_{avg} = v_A + \frac{1}{2}(a(B-A)) = v_A + a\frac{(B-A)}{2}[/tex]

which is the speed at time (B-A)/2.

AM
 
  • #7
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Thanks Andrew.

hk
 
  • #8
lightgrav
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d/v certainly has UNITS of time,
but you really NEED to specify what distance you mean
(midpoint? location at mid-time? total distance?)
and what velocity you mean
(slowest=v_A ? fastest=v_B ? average? v at mid-point?)

time-average of velocity = total displacement/total duration
 

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