Help with Questions: Circles, Vectors, Wedges & dv/dt=g-kv

[mcintyre_ie]

Hey,
I’d appreciate some help with these questions:

(c) Find the centre and radius of the circle x^2 + y^2 – x – y – 12 = 0.
Find the equations of the tangent to this circle which are parallel to the line 7x – y = 0.

Ok, so I found the line has a slope of 7, and the circle has a centre at (½ , ½) and a radius of 5/root2

Using this info, I said that the line equation is y = 7x + c and 7x – y + c = 0. The distance from (½,½) to 7x – y + c = 0 is 5/root2, and used the perpendicular distance from a point to a line formula to get a quadratic equation for C. But, I got some pretty weird answers (c = 14.18033989 and c = -8.180339887). These don’t seem right, since the answer’s aren’t supposed to go into decimals… Any ideas on where I’ve gone wrong?

(c) o, x, y and z are non-collinear vectors where o is the origin, show that:
(i) x.y = y.x
(ii) x.(y + z) = x.y + x.z

On this one, I’m totally stumped. I don’t know where to start in proving them. Any advice?

Q.4 A smooth wedge of mass 4m and slope 45 degrees is placed on a smooth horizontal surface. A particle of mass m is placed on the inclined face of the wedge. The system is released from rest.
(i) Find the speed of the mass relative to the wedge when the speed of the wedge is ½ m./s.

Ok, so I got an acceleration of 49/45 m/s^2 for the wedge and 382/[45(root2)] for the particle, used the formula v = u + at, where v = ½, u = 0, and a = (49/45) to get a time t = 45/98. I then used this t value in the equation v = u + at, getting an answer of (root8) m/s. Does that sound right?

Find the general solution to:
dv/dt = g – kv,
where g and k are constants. Hence, show that lim v (as t goes to infinity) = g/k.

Ok, so I think I got the first section right ( -1/k.ln[g – kv] = t + c ), but I’m unsure how to proceed.

Any help is appreciated.

 

[arildno]

I found the radius in c) to be 7/2 (r^(2)=1/4+12=49/4)

 

[arildno]

As for the second c) (about non-collinear vectors), use the definition of the innerproduct, and assume that:
x=(x1,x2,x3), y=(y1,y2,y3), z=(z1,z2,z3).

As to your general solution, multiply the equation with -k, and exponentiate the equation to get rid of the logarithm.
Rearranging yields the result.

 

[mcintyre_ie]

I found the radius in c) to be 7/2 (r^(2)=1/4+12=49/4)

I found it to be r^2 = 1/4 + 1/4 + 12.

As for the second c) (about non-collinear vectors), use the definition of the innerproduct, and assume that:
x=(x1,x2,x3), y=(y1,y2,y3), z=(z1,z2,z3).

We don't have the innerproduct on our course, only dot product and scalar product, so it has to be solved using either or both of those (or other *basic* rules.

As to your general solution, multiply the equation with -k, and exponentiate the equation to get rid of the logarithm.
Rearranging yields the result.

Ok, I'll try it, but I'm not so sure about the lim part.

Thanks for the help

 

[arildno]

1. You're absolutely right about the radius.
2. The inner product is the same as the scalar product

 

[arildno]

On the scalar product, I'll do the first:
x.y=x1*y1+x2*y2+x3*y3=y1*x1+y2*x2+y3*x3=y.x, as required