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)  (1)/(8)int_(0)^(8)a(t)dt  (x(8)-x(0))/(8)  int_(0)^(8)v(t)dt

Bytelearn · Accepted Answer

Calculate Average Velocity: To find the average velocity of the particle over the interval from t = 0 to t = 8, we need to use the definition of average velocity, which is the total displacement divided by the total time. The displacement is the change in position, so we can use the formula:  Average velocity = (x(8) - x(0)) / (8 - 0)  We need to find the values of x(8) and x(0) to calculate the average velocity.Use Correct Formula: The problem provides us with different expressions for average velocity, but the correct one to use in this context is the change in position over time, which is given by:  Average velocity = (x(8) - x(0)) / 8  This is because the average velocity is the total displacement (change in position) divided by the total time interval.Express as Integral: We can also express the average velocity as the integral of the velocity function over the time interval divided by the time interval. This is because the integral of the velocity function gives us the net displacement over the time interval. So, we have:  Average velocity = (1/8) * ∫(from 0 to 8) v(t) dt  This integral will give us the value of x(8) - x(0) if we assume that v(t) is the derivative of x(t) with respect to time t.Evaluate Integral: Now, we need to evaluate the integral to find the average velocity. However, the problem does not provide us with the explicit form of the velocity function v(t), so we cannot directly compute the integral. We need the function v(t) or additional information to proceed with the calculation.Unable to Find Value: Since we cannot evaluate the integral without the explicit form of v(t), we cannot find the numerical value of the average velocity. We can only express the average velocity in terms of the integral of v(t) over the interval from 0 to 8, as shown in the previous step.

Full solution

More problems from Find derivatives of sine and cosine functions

Related topics