An object starts from rest and has an acceleration given by \(a=B t^{2}-\frac{1}{2} C t,\) where \(B=2.0 \mathrm{~m} / \mathrm{s}^{4}\) and \(C=-4.0 \mathrm{~m} / \mathrm{s}^{3}\). a) What is the object's velocity after 5.0 s? b) How far has the object moved after \(t=5.0\) s?

The acceleration function is given by: \(a=Bt^2-\frac{1}{2}Ct\) with \(B=2.0\frac{m}{s^4}\) and \(C=-4.0\frac{m}{s^3}\). To find the velocity function, we need to integrate the acceleration with respect to time: \(v(t) = \int a(t) dt = \int (Bt^2-\frac{1}{2}Ct) dt = B\int t^2 dt - \frac{1}{2}C\int t dt \) \(= B(\frac{1}{3}t^3+C_1) - \frac{1}{2}C(\frac{1}{2}t^2 + C_2) = \frac{1}{3}Bt^3 - \frac{1}{4}Ct^2 + C_1\) At \(t=0\), the object starts from rest, so \(v(0)=0\). We can use this to find the constant \(C_1\): \(0 = \frac{1}{3}B(0)^3 - \frac{1}{4}C(0)^2 + C_1 \Rightarrow C_1=0\) Now that we have found the constant, we can write the velocity function as: \(v(t) = \frac{1}{3}Bt^3 - \frac{1}{4}Ct^2\) Now we evaluate the \(v(t)\) for \(t=5.0s\): \(v(5.0s) = \frac{1}{3}(2.0\frac{m}{s^4})(5.0s)^3 - \frac{1}{4}(-4.0\frac{m}{s^3})(5.0s)^2\) \(= 50.0\frac{m}{s}\) The object's velocity at \(t=5.0s\) is \(50.0\frac{m}{s}\).

Now, we need to integrate the velocity function to find the displacement function: \(x(t) = \int v(t) dt = \int (\frac{1}{3}Bt^3 - \frac{1}{4}Ct^2) dt \) \(= \frac{1}{3}B\int t^3 dt - \frac{1}{4}C\int t^2 dt \) \(= \frac{1}{3}B(\frac{1}{4}t^4+C_3) - \frac{1}{4}C(\frac{1}{3}t^3 + C_4) = \frac{1}{12}Bt^4 - \frac{1}{12}Ct^3 + C_4\) At \(t=0\), the object has zero displacement, so \(x(0)=0\). We can use this to find the constant \(C_4\): \(0 = \frac{1}{12}B(0)^4 - \frac{1}{12}C(0)^3 + C_4 \Rightarrow C_4=0\) Now that we have found the constant, we can write the displacement function as: \(x(t) = \frac{1}{12}Bt^4 - \frac{1}{12}Ct^3\) Now we evaluate the \(x(t)\) for \(t=5.0s\): \(x(5.0s) = \frac{1}{12}(2.0\frac{m}{s^4})(5.0s)^4 - \frac{1}{12}(-4.0\frac{m}{s^3})(5.0s)^3\) \(= 125.0 m\) The object's displacement at \(t=5.0s\) is \(125.0m\). To summarize the results: a) The object's velocity at \(t=5.0s\) is \(50.0\frac{m}{s}\). b) The object's displacement at \(t=5.0s\) is \(125.0m\).