Final answer:
To find the position of the particle, integrate the acceleration function twice. First, integrate a(t) to get v(t), then integrate v(t) to get s(t). The position function is s(t) = (1/3)t^3 - (7/2)t^2 - 3t + 4.
Step-by-step explanation:
To find the position of the particle, we need to integrate the acceleration function twice. Given that a(t) = 2t - 7, we integrate it once to get the velocity function v(t), and then integrate v(t) to get the position function s(t).
Integrating a(t) with respect to t gives us v(t) = t^2 - 7t + C, where C is the constant of integration. Since the particle's velocity at t=0 is -3, we can substitute t=0 and v= -3 into the velocity function to solve for the constant C. This gives us -3 = 0^2 - 7(0) + C, which simplifies to C = -3.
Substituting C = -3 back into the velocity function, we have v(t) = t^2 - 7t - 3. Finally, we integrate v(t) with respect to t to find the position function s(t). Integrating v(t) gives us s(t) = (1/3)t^3 - (7/2)t^2 - 3t + D, where D is the constant of integration. Since the particle's position at t=0 is 4, we can substitute t=0 and s=4 into the position function to solve for the constant D. This gives us 4 = (1/3)(0)^3 - (7/2)(0)^2 - 3(0) + D, which simplifies to D = 4.
Substituting D = 4 back into the position function, we have s(t) = (1/3)t^3 - (7/2)t^2 - 3t + 4.