Final answer:
To find the particle's velocity and position at various times, we need to integrate the given acceleration function. The integral of the acceleration provides the velocity, and the integral of the velocity provides the position. By applying these methods and using the initial conditions provided, we can determine the expressions for the particle’s velocity and position at any given time.
Step-by-step explanation:
The subject of this question is the movement of a particle along the x-axis, described by its acceleration function, initial position, and initial velocity. So, in order to find the velocity and position of the particle at any given time t, we would need to integrate the acceleration function.
Focusing on the provided acceleration function, a(t)=3/t², we know that to find the velocity at a certain time, we will need to find the integral of this function. You might know that the integral of a(t) with respect to t is v(t), or velocity. For example, if a(t) = 3/t², the antiderivative would be v(t) = -3/t + C₁, where C₁ is the initial velocity. It's given that when t = 1, the particle has a velocity of 2, so C₁ becomes 2. Thus, v(t) = -3/t + 2.
Now, to find x(t), the position of the particle at time t, we need to integrate the velocity function v(t). Hence, the antiderivative of v(t) would be x(t) = 3ln|t| + 2t + C₂, where C₂ is the initial position of the particle. Given that at t = 1, the position of the particle is 6, our C₂ will be 1. Therefore, x(t) = 3ln|t| + 2t + 1.
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