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Vector Calculations and Forces - Physics Problems Solved

Solve vector problems involving forces, velocities, and angles with step-by-step explanations. Find resultant, magnitude, and direction of vectors.

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Vector Calculations and Forces - Physics Problems Solved

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  1. 10) A canoe is paddling West at 5 mph across a river that is North-South and has a southern current running 7 mph. Find the resultant and magnitude of the canoe as it paddles across the river. r = < x + x , y + y > < - 5 , 0 > + < 0 , -7 > < -5 , -7 > <-5,0> 5 W r = 7 <0,-7> mag = √ x2 + y2 = √ (-5)2 + (-7)2 = 8.6 mph

  2. 11) A plane is flying at a heading of (navigation) 230° at a speed of 500 mph, when it encounters a wind blowing due East at 25 mph. What is the resultant and magnitude of the plane when you take into account the wind? r = < x + x , y + y > < 25 , 0 > + < -383.0 , -321.4 > < -358.0 , -321.4 > <25,0> 25 r = W 230 500 <-383.0,-321.4> mag = √ x2 + y2 < mag cos θ , mag sin θ > = √ (-358)2 + (-321.4)2 = 481.1 mph < 500 cos 40, 500 sin 40 > < 383.0 , 321.4 > Quad III = < -383.0 , -321.4 >

  3. 12) A force of 100 lbs. is needed to pull a boy riding in a wagon (going East). The force is applied to the handle which is at a 40° angle to the horizontal. What is the force of gravity on the wagon? [gravity goes straight down] The 100 lbs of force is the resultant. We need to find the downward vector that creates that force. < 76.6 , 64.3 > 100 40 W < ? , 0 > r = < x + x , y + y > < 0, ? > < ? , 0 > East + < 0 , ? > South < 76.6 , 64.3 > < mag cos θ , mag sin θ > r = < 100 cos 40, 100 sin 40 > < 76.6 , 64.3 > Quad I South force is 64.3 lbs.

  4. 13) Find the component vector of a submarine that is diving down at an angle of 35° to the horizon at a speed of 12 knots. Assume the submarine is diving in a south easternly direction. < mag cos θ , mag sin θ > E 35 < 12 cos 35, 12 sin 35 > < 9.8 , 6.9 > Quad IV = < 9.8 , -6.9 > 12 <x,y>

  5. 14) Three dogs are playing tug of war with a long rope. One dog is pulling North with 45 lbs of force. Another dog is pulling SW with 35 lbs of force. And the last dog is pulling due East with 20 lbs of force. Find the resultant, magnitude & angle of the rope’s travel. <0,45> r = < x + x + x , y + y + y > < 0 , 45 > < 20 , 0 > + < -24.7 , -24.7 > < -4.7 , 20.3 > 45 20 W 45 r = <20,0> 35 mag = √ x2 + y2 <-24.7,-24.7> = √ (-4.7)2 + (20.3)2 = 20.8 lbs. < mag cos θ , mag sin θ > Angle = tan-1(y/x) < 35cos 45, 35sin 45 > < 24.7 , 24.7 > Quad III = < -24.7 , -24.7 > tan-1(20.3/-4.7) = -76.96 r = Quad II so add 180. -77.0 + 180 = 103.0

  6. 15) Two boys are blowing on a small toy boat in the bathtub. One boy is blowing on it at an angle of S 40° W with a force of 12. The other boy is blowing with a force of 15 at an angle of S 70°E. Find the resultant and magnitude the toy boat will travel. < mag cos θ , mag sin θ > < 15 cos 20, 15 sin 20 > < 14.1 , 5.1 > Quad IV < 14.1 , -5.1 > W E 20 50 15 70 40 12 < 12 cos 50, 12 sin 50 > < 7.7 , 9.2 > Quad III < -7.7 , -9.2 > r = < x + x , y + y > < 14.1 , -5.1 > + < -7.7 , -9.2 > = < 6.4 , -14.3 > mag = √ x2 + y2 = √ (6.4)2 + (-14.3)2 = 15.7 force

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