Not sure if anyone has interest in this type of problem. My Physics textbook (which I am self-studying) has a section for "Computer Exercises", where the intent is not that you work the problems out by hand, but write code to approximate the answer for you. Unfortunately, the answers for these specific questions aren't provided in the book. I've written the code to do the approximations, summing values over minute changes in time, which I'm fairly confident is correct. Just looking to see if someone else can verify my answers.

Problem 1: The acceleration of a particle varies as a function of it's position, as described in the following equation:

a = a0 * e ^ (-b * x)

Where a0 = 3 (units of m/s^2) and b = 1 (units of m^-1)

Use numerical integration to find the position and velocity of the particle at 2.37 seconds.

Edit to add: it is given that x(0) = 0 m and v(0) = 0 m/s

I get v(2.37) ~ 2.435 m/s

x(2.37) ~ 4.425 m

Problem 2: The acceleration (in m/s^2) of a particle varies with time (in s), as described in the following equation:

a = (3 + t^3) ^ 0.5

Use numerical integration to find the position and velocity of the particle at 5.7 seconds.

Edit to add: it is given that x(0) = 0 m and v(0) = 0 m/s

I get v(5.7) ~ 33.97 m/s

x(5.7) ~ 62.83 m