Final answer:
The position of a particle moving along the x-axis can be found using the formula x(t) = x0 + v0t + 0.5at^2. In this case, we are given the acceleration function a(t) = 2 cos(t) − 7 sin(t), the initial position x(0) = 0, and the initial velocity v(0) = 8. By integrating the acceleration function to find the velocity function and then integrating the velocity function to find the position function, we can determine the position of the particle at a specific time.
Therefore, the position function of the particle is s(t) = -2 cos(t) + 7 sin(t) + 2.
Step-by-step explanation:
The position of a particle moving along the x-axis can be found using the formula x(t) = x0 + v0t + 0.5at^2, where x(t) is the position at time t, x0 is the initial position, v0 is the initial velocity, a is the acceleration, and t is the time. In this case, we are given the acceleration function a(t) = 2 cos(t) − 7 sin(t), the initial position x(0) = 0, and the initial velocity v(0) = 8.
To find the position of the particle at a specific time, we need to integrate the acceleration function to find the velocity function, and then integrate the velocity function to find the position function.
First, let's find the velocity function v(t) by integrating a(t). The indefinite integral of 2 cos(t) is 2 sin(t), and the indefinite integral of -7 sin(t) is 7 cos(t). Therefore, v(t) = 2 sin(t) + 7 cos(t).
Next, let's find the position function s(t) by integrating v(t). The indefinite integral of 2 sin(t) is -2 cos(t), and the indefinite integral of 7 cos(t) is 7 sin(t). Therefore, s(t) = -2 cos(t) + 7 sin(t) + C, where C is the constant of integration.
To find the value of the constant of integration C, we can use the initial condition s(0) = 0. Plugging in t = 0 and s(0) = 0 into the position function, we get 0 = -2 cos(0) + 7 sin(0) + C. Simplifying this equation, we find C = 2.
Therefore, the position function of the particle is s(t) = -2 cos(t) + 7 sin(t) + 2.