Quiz4U! Find the resulting velocity vector?

Question

Suppose that a ball in 2D with an incoming velocity vector of [8 -6] bounces on a wall defined by the points (7,1) and (8,6). Find its resulting velocity vector after it hits the wall.

Hint: First find the boundary wall vector B=[8-7 6-1] = [1 5]. Thus, the normal vector is

N=[5 -1]. Then, the normalized vector is N' = N/||N|= [.98 -.2].

Possible Answers:

[-3.42 -9.40]

[-9.72 -2.38]

[-9.85 -1.74]

[-7.66 -6.43]

[-4.85 -8.75]

Answer 1

I don't get any of those answers. I get (-9.69, -2.46) (approx.) My calculations are as follows:

As you state, a vector for the wall is (1, 5).

The angle of reflection must equal the angle of incidence.

Also, I'm assuming that the ball has the same speed after the collision, i.e.

x^2 + y^2 = 8^2 + (-6)^2

= 64 + 36 = 100

It follows that the dot product of the wall vector and the resulting velocity vector is equal to the dot product of the wall vector and incoming velocity vector.

(1, 5).(8, -6) = 8 - 30 = - 22

So we require (x, y) s.t. (x, y).(1, 5) = - 22

i.e. x + 5y = - 22

So we need to solve simultaneously

x + 5y = - 22 and

x^2 + y^2 = 100

Leaving out my working, I get:

(x, y) = (8, -6) (as expected) or

(-126/13, -32/13)

= (-9.69, -2.46) (approx.)

I have drawn a diagram and it looks right.