resultant-speed angle-of-projection simultaneously-velocity

Question

1. a ship is 100km north west of cape town haboyr and its sails directlt north at a velocity of 30 km/h in eight hours. Calculate its resultant speed with reference to the cape town harbour in magnitude and direction. 2. calculate the angle of projection of a projectile in order to hit a target 12km away. The initial velocity is 700m/s and the time of the flight in 80 seconds. 3. two vehicles start simultaneously at a fork in a road. Vehicles M travels north west at 120km/h and vehicle N travels east at 142 km/h. calculate the velocity of vehicle M relative to the velocity of vehicle N in magnitude and direction.

Answer 1

1)

Ship is 100 km north west of cape town .

Ship took 8 hours to reach the north west point .

The velocity of the ship while reaching the point is 100 km / 8 = 12.5 km/hr

Ship sails to north at a velocity 30 km/hr .

The resultant velocity of the ship can be determined by using the following the below diagram .

So the resultant velocity with respect to cape town can be calculated using cosine rule .

c² = 30² + 12.5² -2(30)(12.5) cos(135°) 

c² = 900 + 156.25 + 530.33

c² = 1586.58

c = 39.83 km/hr 

The resultant velocity of ship with respect to cape town is 39.83 km/hr .

 The direction of ship with respect to cape town can be calculated using sine rule .

sin A / 30  = 0.7071 / 39.83

sin A =  0.017753 * 30

sin A = 0.53259

A = arcsin ( 0.53259)

A = 32.18°

 The direction of ship with respect to cape town is 32.18° .

Answer 2

2)

The initial velocity of the object is (v₀) 700 m/s .

Target distance is 12 km .

Earth gravity g = 9.8 m/s .

We can evaluate the angle of projection  with the following formula 

Maximum distance ( target distance ) =  (v₀²sin2θ)/g 

12000 = [(700)² sin2θ ] /9.8

(700)² sin2θ = 12000*9.8

sin2θ = 117600 / 700² 

sin2θ =  117600 / 490000

sin2θ = 0.24

2θ = arcsin ( 0.24)

2θ = 13.886

θ = 13.886 /2

θ = 6.943

So the angle of projection is θ = 6.943° .

Answer 3

Vehicles M  travels at 120 km/hr to north west .

vehicle N  travels at 142 km/h in a direction of east .

So the relative velocity  can be determined by the below diagram .

So the relative velocity can be calculated using cosine rule .

c² = 120² + 142² -2(120)(142) cos(135°) 

c² = 14400 +20164 + 24098.199

c² = 58662.199

c = 242.202

The relative velocity  is 242.202 km/hr .

 The direction of two vehicles can be calculated using sine rule .

sin A / 142  = sin 135 / 242.202

sin A =  0.002919 * 142

sin A = 0.4145

A = arcsin ( 0.4145)

A = 24.49°

 The direction of two vehicles is 24.49° .