A particle moves along the x-axis with position function xt) = sint -.
_t for 0 St < r. What is the total
2
distance the particle travels on that time interval?
To find the total distance traveled by the particle, we need to calculate the sum of the absolute values of the distances traveled in each direction.
The direction of the particle's motion can change when the velocity changes sign. In this case, the velocity function is given by v(t) = cos(t) - 2t. To find the points where the velocity changes sign, we solve the equation cos(t) - 2t = 0.
cos(t) - 2t = 0
2t = cos(t)
t = cos(t)/2
We can approximate the positive solution of this equation using numerical methods, such as Newton's method, to find t ≈ 0.4502.
The particle is initially moving in the positive x-direction, so the distance traveled in that direction is the integral of the absolute value of the velocity function from 0 to the positive solution t ≈ 0.4502.
∫[0 to t ≈ 0.4502] |cos(t) - 2t| dt
To evaluate this integral, we can divide the interval [0, t ≈ 0.4502] into subintervals where the velocity function is either positive or negative. The absolute value of the velocity function is equal to the velocity function when the velocity is positive, so we can rewrite the integral as:
∫[0 to t ≈ 0.4502] (cos(t) - 2t) dt
Evaluating this integral gives us the distance traveled in the positive x-direction.
Next, the particle changes direction and starts moving in the negative x-direction. To find the distance traveled in this direction, we need to calculate the integral of the absolute value of the velocity function from the positive solution t ≈ 0.4502 to the maximum time interval t = r = 2.
∫[t ≈ 0.4502 to 2] |cos(t) - 2t| dt
Again, we can divide the interval [t ≈ 0.4502, 2] into subintervals where the velocity function is either positive or negative. The absolute value of the velocity function is equal to the negation of the velocity function when the velocity is negative, so we can rewrite the integral as:
∫[t ≈ 0.4502 to 2] -(cos(t) - 2t) dt
Evaluating this integral gives us the distance traveled in the negative x-direction.
Finally, we can find the total distance traveled by summing the distances traveled in each direction:
Total distance = Distance traveled in positive x-direction + Distance traveled in negative x-direction.
Note that the exact values of the integrals and the numerical approximation of t ≈ 0.4502 would depend on the specific values of r and the precision desired in the calculations.