Homework(MAthematics)

Classical Mechanics 1 Name MA-142 Summative Assignment SA-1

Number

Submit through Canvas by 19 February 11am

Tutor

Feedback Mark /10 Date Marked / /2021 Presentation

Marking Criteria: Full marks will be awarded for work that 1) is mathe- matically correct, 2) shows an understanding of material presented in lectures, 3) gives details of all calculations and reasoning, and 4) is presented in a logical and clear manner.

1. Let i,j,k be a standard basis. Consider the vectors

a = 2i + 4j + 2k, b = i−j − 2k.

(i) Find a ·b and use it to find the angle α between a and b. (Hint: Recall that cos−1(−x) = π − cos−1 x for any x.)

[1 Mark]

(ii) Find a vector n which is orthogonal to both a and b and has the magnitude 6.

[1 Mark]

(iii) Let an origin be fixed. Let p be the plane which is parallel to both vectors a and b and which passes through the point C(1, 2,−2). Find the distance from the origin to the plane p.

(Hint: Recall that a vector is orthogonal to a plane if it is orthogonal to two vectors which are parallel to that plane and not parallel to each other.)

[1 Mark]

(iv) Consider the point D(1,−4, 1). Let q be the plane which is passes through the origin and is parallel to p. Let E be the point of the intersection of the line CD and the plane q. Find E.

(Hint: write down the vector equation of q, note that parallel planes share the same normal vectors.)

[2 Marks]

– 1 – Turn over.

2. A bee flies in a path so that its velocity vector v = v(t) at time t ≥ 0 is given by

v = (6t + 1)i + (8t− 7)j − 3k.

Let r = r(t) denote the position vector of the bee at time t. Suppose that, initially, at the moment of time t = 0, the bee was at the position

r(0) = 4i− 3j + 2k.

(i) Find r(t).

[1 Mark]

(ii) At which time t > 0 the position vector r is orthogonal to the accel- eration vector a = a(t)?

[1 Mark]

(iii) For which values of p ∈ R the vector

b = r − tv + pt2a

is a constant vector in t (i.e. b does not depend on t)?

[1 Mark]

(iv) At which time t > 0 the acceleration a does not have a tangential component? Find the curvature of the trajectory at this time t.

(Hint: Find v(t) = |v(t)| and use the formula for acceleration’s compo- nents.)

[2 Marks]

– 2 –