Answer to Question #193867 in Math for Ola

Question

A cart travels continuously around a circular track with a constant acceleration. The track has a length of 10km.

At a specific time, t = 0, its position is measured.

It takes half an hour for the cart to complete one full lap of the track.

It takes two hours (from time t = 0) for the cart to complete ten full laps of the track. The following formula describes this situation

s = u0t + a t^2 /2

   where

• s is the displacement, measured in km

• u0 is the velocity of the cart when t = 0, measured in kmh-1

• t is the time taken, measured in hours

• a is the acceleration of the cart, measured in kmh-2

a) Use the information about the position of the cart after half an hour to write a formula linking u0 and a.

b) Use the information about the position of the cart after two hours to write another formula linking u0 and a.

c) Use your equations to find the initial velocity of the cart and its acceleration.

   

Answer 1

$$s=u_0t+\dfrac{at^2}{2}$$

$$l=10\ km$$

a)

$$s=10=u_0(\dfrac{1}{2})+\dfrac{a(\dfrac{1}{2})^2}{2}$$

$$4u_0+a=80$$

b)

$$s=10\cdot10=u_0(2)+\dfrac{a(2)^2}{2}$$

$$u_0+a=50$$

c)

$$\begin{matrix} 4u_0+a=80 \\ u_0+a=50 \end{matrix}$$

$$\begin{matrix} 4u_0+a-(u_0+a)=80-50 \\ u_0+a-u_0=50-u_0 \end{matrix}$$

$$\begin{matrix} 3u_0=30 \\ a=50-u_0 \end{matrix}$$

$$\begin{matrix} u_0=10 \ kmh^{-1} \\ a=40\ km h^{-2} \end{matrix}$$