Solve Applied Maths Urgent Probl. 1: 20 Marks

[bilal98732]

Problem 1 20 marks (i) ˆx = (1, 0) and ˆy = (0, 1) are the Cartesian unit vectors and the vectors v1 and v2 are defined as v1 = −4ˆx + 0ˆy , v2 = 2ˆx − 7ˆy . Determine the polar coordinate unit vectors ˆr and ˆθ for v1 and v2 and hence express v1 and v2 as a linear combination of ˆr and ˆθ. [4]

(ii) A particle’s motion is described by the following position vector r(t) = αtxˆ + (βt2 − t)ˆy Determine the polar coordinate unit vectors ˆr and ˆθ for r. [4]

(iii) By differentiating with respect to time r(t), given in (ii) show that the velocity vector written in a Cartesian basis for this particle is v(t) = αxˆ + (2βt − 1)ˆy . [2] (iv) Using the ˆr and ˆθ you found in (ii) above, write v(t) as a linear combination of rˆ and ˆθ. [4]

(v) Differentiate the expression for r(t) you got in part (ii) (in terms of ˆr and ˆθ, and using the expressions ˙rˆ = ˙θ ˆθ , ˙ˆθ = − ˙θrˆ derived in the lectures, show that you obtain the same answer as in part (iv) [6]where r(vector) = r modulus * r
r^ = cos(pheta)x + sin(pheta)y
pheta^ = -sin(pheta)x +cos(pheta)yi am able to do i) and ii) but unbale to expres v(t) in terms of r and pheta

 

[RUber]

Problem 1 20 marks (i) ˆx = (1, 0) and ˆy = (0, 1) are the Cartesian unit vectors and the vectors v1 and v2 are defined as v1 = −4ˆx + 0ˆy , v2 = 2ˆx − 7ˆy . Determine the polar coordinate unit vectors ˆr and ˆθ for v1 and v2 and hence express v1 and v2 as a linear combination of ˆr and ˆθ. [4]

(ii) A particle’s motion is described by the following position vector r(t) = αtxˆ + (βt2 − t)ˆy Determine the polar coordinate unit vectors ˆr and ˆθ for r. [4]

What did you get for the position vector?
What do you get when you take the time derivative of that expression?