A particle moves along the x-axis such that at any time t = 0 its position is x(t), its velocity is v(t), and its acceleration is a(t). What is the average velocity of the particle on the interval 0

Question

A particle moves along the  x-axis such that at any time  t >= 0 its position is  x(t), its velocity is  v(t), and its acceleration is  a(t). What is the average velocity of the particle on the interval  0 <= t <= 8 ?  (v(8)-v(0))/(8)  (a(8)-a(0))/(8)  (1)/(8)int_(0)^(8)v(t)dt  (1)/(8)int_(0)^(8)a(t)dt

Bytelearn · Accepted Answer

Understand Average Velocity: Understand the definition of average velocity. The average velocity of a particle over a time interval is defined as the total displacement divided by the total time taken. In mathematical terms, if x(t) is the position function, then the average velocity $ \bar{v} $ over the interval [a, b] is given by: $$ \bar{v} = \frac{x(b) - x(a)}{b - a} $$Identify Correct Formula: Identify the correct formula to use. From the options given, the correct formula to calculate the average velocity is the one that involves the position function x(t), not the velocity v(t) or the acceleration a(t). Therefore, the correct formula is: $$ \bar{v} = \frac{1}{8} \int_{0}^{8} v(t) \, dt $$ This is because the integral of the velocity function over the interval [0, 8] gives the total displacement, and dividing by the time interval (8 - 0 = 8) gives the average velocity.Choose Correct Answer: Choose the correct answer based on the formula. The correct answer is the one that matches the formula for average velocity using the integral of the velocity function: $$ \bar{v} = \frac{1}{8} \int_{0}^{8} v(t) \, dt $$

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