Displacement from velocity A particle moves along a line with a velocity given by v(t) = 5 sin πt, starting with an initial position s(0) = 0 . Find the displacement of the particle between t = 0 and t = 2 , which is given by s(t) = ∫₀² v(t) dt . Find the distance traveled by the particle during this interval, which is ∫₀² |v(t)| dt .

Verified step by step guidance

1. Step 1: Understand the problem. The displacement of the particle is calculated using the definite integral of the velocity function v(t) over the interval [0, 2]. The distance traveled is calculated using the definite integral of the absolute value of the velocity function |v(t)| over the same interval.
2. Step 2: Write the displacement formula. The displacement is given by s(t) = ∫₀² v(t) dt. Substitute v(t) = 5 sin(πt) into the integral: s(t) = ∫₀² 5 sin(πt) dt.
3. Step 3: Solve the integral for displacement. Use the integral rule for sine: ∫ sin(ax) dx = -(1/a) cos(ax) + C. Here, a = π, so the integral becomes ∫₀² 5 sin(πt) dt = -5/π [cos(πt)] evaluated from t = 0 to t = 2.
4. Step 4: Write the formula for distance traveled. The distance is given by ∫₀² |v(t)| dt. Since v(t) = 5 sin(πt), the absolute value |v(t)| must be considered. Analyze the behavior of sin(πt) over [0, 2] to determine where it is positive or negative, and split the integral accordingly.
5. Step 5: Solve the integral for distance. Break the integral into intervals where sin(πt) is positive and negative. For t ∈ [0, 1], sin(πt) ≥ 0, so |v(t)| = 5 sin(πt). For t ∈ [1, 2], sin(πt) < 0, so |v(t)| = -5 sin(πt). Compute ∫₀¹ 5 sin(πt) dt and ∫₁² -5 sin(πt) dt separately, then add the results to find the total distance traveled.

Key Concepts

Velocity and Displacement

Velocity is the rate of change of position with respect to time, represented mathematically as v(t). Displacement refers to the change in position of a particle over a specific time interval, calculated by integrating the velocity function. In this case, the displacement is found by evaluating the integral of v(t) from t = 0 to t = 2.

Definite Integral

A definite integral calculates the accumulation of quantities, such as area under a curve, over a specified interval. It is denoted as ∫ₐᵇ f(x) dx, where a and b are the limits of integration. In the context of this problem, the definite integral of the velocity function gives the displacement of the particle over the interval from t = 0 to t = 2.

Absolute Value of Velocity

The absolute value of velocity, |v(t)|, represents the speed of the particle regardless of direction. When calculating the total distance traveled, it is essential to integrate the absolute value of the velocity function over the given interval. This ensures that any changes in direction do not cancel out the distance covered, providing a true measure of the path length.