Question

A snail, an ant, and a beetle are having a race. The course is a straight line from position p subscript s equals open square brackets table row 2 row 0 row 3 end table close square brackets to position p subscript f equals open square brackets table row 6 row 0 row cell negative 5 end cell end table close square brackets. The ant finishes the race in 2 seconds. Write down the parametric equation for the ant's position as a function of time, A left parenthesis t right parenthesis, assuming that the race starts at time t equals 0.

Answer 1

We have the given scenario as follows:

1. Initial position is given by ps = [2,0,3]
2. Final position is given by pf = [6,0,-5]
3. Total time taken by the ant to move from starting position to the final position = 2 seconds

Few important points to be noted are:

• We have been given a 3D cordinate plane since 3 dimensions are specified for the position.
• The intial position is given at time t=0.

Considering the above two points in mind, we can write the required equation as follows:

Equation for position at time t=0 is given as:

A(0) = 2.î + 0.ĵ + 3.k̂

Firstly, lets consider how the coordinate position changes in unit time i.e, per second for which we need to consider the change that has occured in each coordinate x,y and z.

Change in x-coordinate per unit time = 6-2/2 = 4/2= 2 units

Change in y-coordinate per unit time = 0-0/2 = 0/2= 0 unit

Change in z-coordinate per unit time = -5-3/2 = -8/2= -4 units

So, using the above difference per unit time and multiplying each term by time variable"t" then placing them in the respective places, we have the required equation as a function of time as follows:

A(t) = (2+2.t).î + (0+0.t).ĵ + (3-4.t).k̂

We can also check the correctness of the equation by substituting for t=0 and t=2

A(0) = (2+2.0).î + (0+0.0).ĵ + (3-4.0).k̂ = 2.î + 0.ĵ + 3.k̂ = ps

A(2) = (2+2.2).î + (0+0.2).ĵ + (3-4.2).k̂ = (2+4).î + (0+0).ĵ + (3-8).k̂ = 6.î + 0.ĵ - 5.k̂ = pf

Hence, we can see that above equation is correct.

(Thank You!)