A particle travels along the x-axis such that its velocity is given by v(t)=(t^(2.3)+5)sin(2t). What is the distance traveled by the particle over the interval 0

Question

A particle travels along the  x-axis such that its velocity is given by  v(t)=(t^(2.3)+5)sin(2t). What is the distance traveled by the particle over the interval  0 <= t <= 5 ? You may use a calculator and round your answer to the nearest thousandth. Answer:

Bytelearn · Accepted Answer

Set up integral: To find the distance traveled by the particle, we need to integrate the absolute value of the velocity function over the given interval. The absolute value is necessary because velocity can be negative, which would indicate the particle is moving in the opposite direction, but distance is always positive.Find intervals: First, we set up the integral of the absolute value of the velocity function from t = 0 to t = 5. Distance = ∫ from 0 to 5 |v(t)| dt = ∫ from 0 to 5 |(t^(2.3) + 5)sin(2t)| dtLocate zeros: Since we cannot directly integrate the absolute value of a function without knowing where the function is positive or negative, we need to find the intervals where the velocity function v(t) = (t^(2.3) + 5)sin(2t) is positive or negative between 0 and 5.Break into pieces: We look for the zeros of the velocity function within the interval [0, 5] to determine where the function changes sign. The zeros of sin(2t) occur at t = k*pi/2 for k = 0, 1, 2, ..., where k is an integer. Within the interval [0, 5], this happens at t = 0, pi/2, pi, 3pi/2, and 2pi (approximately 0, 1.57, 3.14, 4.71, and 6.28). However, since we are only considering the interval from 0 to 5, we will only consider the zeros at t = 0, pi/2, pi, and 3pi/2.Integrate piecewise: We now break the integral into pieces where the velocity function does not change sign. We will integrate from 0 to pi/2, pi/2 to pi, and pi to 3pi/2, taking the absolute value of the velocity function in each interval.Calculate total distance: We calculate the integral piecewise, using a calculator to evaluate the definite integrals and summing them up to find the total distance. Distance = ∫ from 0 to pi/2 |(t^(2.3) + 5)sin(2t)| dt + ∫ from pi/2 to pi |(t^(2.3) + 5)sin(2t)| dt + ∫ from pi to 3pi/2 |(t^(2.3) + 5)sin(2t)| dtEvaluate integrals: Using a calculator, we evaluate each integral and sum the absolute values to find the total distance. This step involves numerical computation and rounding to the nearest thousandth as instructed.Find total distance: After performing the calculations, we find the total distance traveled by the particle over the interval from 0 to 5. Let's assume the calculator gives us a value of D (since we cannot actually compute it here).

A particle travels along the $x$ -axis such that its velocity is given by $v(t)=\left(t^{2.3}+5\right) \sin (2 t)$ . What is the distance traveled by the particle over the interval $0 \leq t \leq 5$ ? You may use a calculator and round your answer to the nearest thousandth. $\newline$ Answer:

Full solution

More problems from Relate position, velocity, speed, and acceleration using derivatives

Related topics