Why can't i apply the simple average velocity formula?

[rainstom07]

I'm doing a homework problem (i already know the answer) and i came across an error in my logic/application of the formula V_avg = (v + v₀)/2. Hopefully you can help me understand why it's incorrect to use the formula.

x = 12t²-2t³ describes a particle position. the derivative of x = 24t-6t².

The homework question asked me find the average velocity between t = 0 and t = 3.

Using the formula V_avg = Δx/Δt yields 18 m/s... the correct answer.

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When i use the simpler formula: V_avg = (v_f + v_i) / 2 = (x'(3.0)+x'(0.0))/2 = 18/2. I get 9 m/s which is incorrect.

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Adding the velocity at t=3 with the velocity at t=0 and then dividing by 2 should've produced 18 m/s... My logic is clearly wrong, but how?

x' describes the velocity of the particle at t seconds? right?

thanks.

 

[Studiot]

Hello rainstom, welcome to Physics Forums

Your average forumla (t₀+t₃)/2 is only correct if x is a linear function of t ie a straight line.

The function x = 12t²-2t³ is decidedly non linear

The way to derive an average for a non linear function (works for linear as well but is trivial) is to integrate the function and divide by the interval or number of points or samples.

So average = Area/Interval

Does this help and can you now obtain the correct answer?

 

[rainstom07]

Your average forumla (t₀+t₃)/2 is only correct if x is a linear function of t ie a straight line.

Thank you! i knew there was some sort of condition attached to the simpler version of average velocity.

Does this help and can you now obtain the correct answer?

^^ yup

 

[willem2]

Hello rainstom, welcome to Physics Forums

Your average forumla (t₀+t₃)/2 is only correct if x is a linear function of t ie a straight line.

The function x = 12t²-2t³ is decidedly non linear

I suppose you meant v_avg = (v_0 + v_3)/2 is only correct if v is a linear function of t.?

X must then be a quadratic function of t, and x = 12t²-2t³ isn't a quadratic.

 

[Studiot]

Hello willem2 does this attachment help?

The average velocity is the number which if you multiplied it by the time would give you the total distance travelled.