Please offer solutions for questions 1-5

Question 1

An object is traveling along the x axis so that its speed is given by v(l) =tz -5t +6
m/s.

(a) Find the times when the object is at rest.
(b) Find the acceleration at both of these times. lnterpret the acceleration in

terms of the motion of the object, ie whether the object is increasing or
decreasing in speed and in which direction.

(c) lf the object is at x=-1 m at l=1, find the position of the object at time f.
(d) Find the leftmost position of the object for all 80.
(e) Use Excel to graph the position of the object for the first 8 seconds.

Question 2

Two aircraft are approaching an airfield, aircraft A at an altitude of 2,000 feet and
aircraft B at an altitude of 2,500 feet. ln all parts of this question use
measurements in nautical miles for distance and knots for speed. At time l=0
aircraftA is 15 miles due south of the airfield, and is flying due north at a speed
of 12A knots, while aircraft B is 13 miles due east of the airfield and is flying at
1 10 knots.

(a) lf the distance between points A and B in two dimensions is given by
s’=(xo*tr)’+(yn-y)t, then give a formula for the distance in three

dimensions between the points *,a,!a,z,afu:d *u,lu,”u).
(b) State the x, y and z coordinates of each of the aircraft at time f.
(c) Find an expression for the sguare of the distance between the two aircraft.

Expand out the RHS, so that you have a formula of the type
s2 = ot2 + bt + c . Differentiate both sides of this expression.

(d) Solve Ur {= 0, and show that the minimum distance occurs in less than
dt

20 minutes from l=0.
(e) Find the distance at this time.
(Q Find the rate of approach at time f=0.1 hours.

Question 3

lntegrate the following functions.

(a) y =+-zJF +6xcos(3×2) {b) y =x'(zx’-r)ux

Question 4

FiF*.the.,area. bebnpen’ the *ur,v-e,S i, :-1Z;1i

2

,. ar11fl’,,’yr414,..’;?fding the :
.pojtl$ of.interse-digL.,algeb.raiel tvle!! ure,yOu-draw,a’eket .el,ea y shoring-.. .
the- area’yoqare’,fin j,qg.and-the:p,oi*tsr.of inte a..g ap-h cone.in :
Excel will be perfectly acceptable, as long as you find the points of intersection
algebraically.)

::: ::. :: :: : : , : ,., -t-ttt
– t:=:’

Question 5

Showthatthecubicl=x3+2×2-x-1hasa,”,o,o*erebetwe”n*=o”no
x=1,. Fickesuitable $ta in$:e$g:trh1sdfu f ,,x6;tfum alno.ngst xo=O.a”x g -‘=”-
and x6=g8.and’perfpmq 3,ite.rgtier-,lsd$ l+lr-ethd;,rest.yo.ur..finil eEtimate

…:.: ,:,. :::::::
‘ .:,: :::,:-: .,..:,1

tJ*6: e’airem perfaru.ance.dlata..fsrtlte,’Egt:te.Mark tV contaned,.in,he”tal6i 1″‘t.’.,
questions to answer this question

.:.:. : .. .. . ] :]]:: :
,Estir.nate ths,,f:1g6;;{is.$p.@d,oJ an Edro Mark’tV.aircr,a.t.,at,63 engim.. ,;..
and a weight d 2;800’kg.,at l$A and:e pressure- rr.eight,d 1,{,g$4.feet. ….

2