# Q2s

2010 higher Solutions

Find VAB (5)

Determine the Mag & Direction (5)

Allow one of the vehicles reach the crossroad, intersection

Determine how far the other will be from the Xroad (5)

Draw the Triangle (5)

Determine the shortest distance (5)

So now we know when they are closest to each other

Whats the Time? T1

Well how long did it take for the vehicles to get to the Xroads?

TheShortest Distance / velocity of VAB = T2

T = T1 + T2

Find velocity x the time, to find the distance, then take away from the Intersection (5)

(b)

distance = uT = d (5)

Draw the triangle, s is still important as a side, as will be u and the wind speed

Sin Rule will put us in terms of the angle (5)

Cos2 A + Sin2 A = 1 rearrange

Find an equation for time taken (5)

Express that in u & T (5)

Solutions

Find the Relative Velocity (5)

Establish the Magnitude and the Direction (5) (5)

Draw the Triangle,

Distance / speed = Time (5)

Time x speed of C = Distance (5)

(b)

2009

2.

(a)

Two cars, A and B, travel along two straight roads which intersect at right angles.

A is travelling east at 15 m/s.

B in travelling north at 20 m/s.

At a certain instant both cars are 800 m from the intersection and approaching the intersection.

Find

(i) the shortest distance between the cars

ii) the distance each car is from the intersection when they are nearest to each other.

(b)

The speed of an aeroplane in still air is u km/h.

The aeroplane flies a straight-line course from P to Q, where Q is north of P. If there is no wind blowing the time for the journey from P to Q is T hours.

Find, in terms of u and T,

the time to fly from P to Q if there is a wind blowing from the south-east with a speed of 4 (2)½ km/h.

2008

2.

(a)

Two straight roads cross at right angles.

A woman C, is walking towards the intersection with a uniform speed of 1.5 m/s.

Another woman D is moving towards the intersection with a uniform speed of 2m/s.

C is 100 m away from the intersection as D passes the intersection.

Find

(i) the velocity of C relative D

(ii) the distance of C from the intersection when they are nearest together.

(b) On a particular day the velocity of the wind, in terms of i and j , is xi - 3j where x ∈ N.

and ij are unit vectors in the directions East and North respectively.

To a man travelling due East the wind appears to come from a direction North α° West where tan α = 2.

When he travels due North at the same speed as before, the wind appears to come from a direction North β° West where tan β = 3/2

Find the actual direction of the wind.

2007

2.

(a)

Ship B is travelling west at 24 km/h. Ship A is travelling north at 32 km/h.

At a certain instant ship B is 8 km north-east of ship A.

(i) Find the velocity of ship A relative to ship B.

(ii) Calculate the length of time, to the nearest minute, for which the ships are less than or equal to 8 km apart.

(b)

A man can swim at 3 m/s in still water.

He swims across a river of width 30 metres.

He sets out at an angle of 30° to the bank.

The river flows with a constant speed of 5 m/s parallel to the straight banks.

In crossing the river he is carried downstream a distance d metres. (5)

Now find Tan of alpha put in to the equation (5)

Find Tan of Beta and put into the equation (5)

Get a value for Vw (5)

Decipher the Direction (5)

Find the Relative Velocity (5)

Establish the Magnitude and the Direction (5) (5)

Draw the Triangle,

Distance / speed = Time (5)

Remember that it will be twice the time as it will still be within 8 km

Work out the time (5)

in minutes (5)

(b)

Time to cross = Width / Vi (5)

Distance downstream = Vr - Vj x time to cross (5)

Work it out (5)

2005

2. (a) A woman can swim at u m/s in still water. She swims across a river of width d metres. The river flows with a constant speed of v m/s parallel to the straight banks, where v < u. Crossing the river in the shortest time takes the woman 10 seconds.

Find, in terms of u and v, the time it takes the woman to cross the river by the

shortest path.

(b) Two straight roads intersect at an angle of 45°. Car A is moving towards the

intersection with a uniform speed of p m/s. Car B is moving towards the intersection with a uniform speed of 8 m/s.

The velocity of car A relative to

car B is − 2i −10j , where i j

and are unit perpendicular vectors in the east and north directions, respectively.

At a certain instant car A is 220 2 m from the intersection and car B is 136 m from the intersection.

(i) Find the value of p.

(ii) How far is car A from the intersection at the instant when the cars are

nearest to each other? 