Question

An object moves according to a law of motion, where, its position is described by the following function, s = f(t) = t^4 - 4t + 1. The time t is measured in seconds and s in meter.a. Sketch the velocity graph and determine when is the object moving in the positivedirection.b. Draw a diagram of the motion of the object and determine the total distancetraveled during the first 6 seconds.

Answer 1

(a) The function given is the distance graph. The velocity fuction is the derivative of the distance function. This becomes

`\begin{gathered} f(t)=t^4-4t+1 \\ f^(\prime)(t)=4t^3-4 \\ v=4t^3-4 \end{gathered}`

the graph is shown below

From the velocity graph above, the function is moving in a positive direction from 1 second and beyond

(b) The diagram of motion of the object is shown in the distance graph below

The total distance travelled is the integral of the velocity function from 0 to 6 as shown below

`\begin{gathered} \int_0^64x^3-4 \\ =[(4x^4)/(4)-4x]_0^6 \\ =[x^4-4x]_0^6=6^4-4(6)-(0^4-4(0)) \\ =1296-24-0 \\ =1272 \end{gathered}`

Hence the answer is 1272 m