1.1 Let's say a cart's position is given by x(t) = at³ – Bt. (1) Assume the constants, a and B, are positive. (a) What must be the general type of unit (1like length, mass, time, and combinations of these) for each of the constants, a and B? (b) Draw, by hand, a sketch of the two functions at³ and -ßt for t > 0. Add the two curves graphically to get a final sketch of x(t). (c) Calculate the cart's velocity vr and acceleration az. (d) Find an expression for the time t = to at which the cart is momentarily at rest. Check: If a and ß had the values 1 and 27, respectively (in the correct SI units), you would find t = 3 s. (e) Give the expression for the cart's position x(to) at the time t = to. Always simplify your answer, and confirm that the units are correct.

We are given a function of time which represents the motion of an object. There are two constants with unknown dimensions to find the dimensions of the two constants we have to do the dimension analysis for the entire function.

To calculate the velocity we have to take the first derivative of the function with respect to time.

$\mathit{v}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\frac{\mathbf{d}\mathbf{x}\left(t\right)}{\mathbf{d}\mathbf{t}}$

To calculate acceleration take the second derivative of the given function,

$\mathit{a}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\frac{{\mathbf{d}}^{\mathbf{2}}\left(x\left(t\right)\right)}{\mathbf{d}{\mathbf{t}}^{\mathbf{2}}}$