Forces add as vectors, not as numbers: 30 N and 40 N pushing at right angles make 50 N, not 70. Split each force into its x and y parts, add the parts, and the resultant is the vector those sums make. Its size comes from the Pythagorean theorem and its direction from the inverse tangent. A plane in a wind works the same way: its velocity plus the wind's is the ground velocity, whose size is the ground speed and whose direction, clockwise from north, is the bearing.
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The angle between a 10 lb force and a 60 lb force on a hook is 150∘. Find the magnitude of their resultant and the angle it makes with the 10 lb force.
Two ropes tied to a post pull with 50 lb and 55 lb, and the angle between the ropes is 45∘. Find the size of the resultant and its angle from the 50 lb rope.
Forces of 80 N and 35 N act on an object with an angle of 45∘ between them. Find the size of the resultant and its angle from the 80 N force.
Two people push a cart, one with 10 lb and one with 85 lb, their pushes 100∘ apart. Find the size of the resultant and its angle from the 10 lb push.
Two ropes tied to a post pull with 75 N and 40 N, and the angle between the ropes is 80∘. Find the size of the resultant and its angle from the 75 N rope.
Two people push a cart, one with 35 N and one with 50 N, their pushes 135∘ apart. Find the size of the resultant and its angle from the 35 N push.
The angle between a 95 N force and a 50 N force on a hook is 80∘. Find the magnitude of their resultant and the angle it makes with the 95 N force.
Forces of 80 lb and 15 lb act on an object with an angle of 150∘ between them. Find the size of the resultant and its angle from the 80 lb force.
The angle between a 45 N force and an 85 N force on a hook is 50∘. Find the magnitude of their resultant and the angle it makes with the 45 N force.
Forces of 35 lb and 70 lb act on an object with an angle of 135∘ between them. Find the size of the resultant and its angle from the 35 lb force.
Find ground speed and bearing with wind
A plane flies at 310 mph on a bearing of 220∘. The wind blows 70 mph toward the south. Find the ground speed and the bearing, to the nearest tenth.
A plane flies at 300 mph on a bearing of 060∘. The wind blows 50 mph toward the north. Find the ground speed and the bearing, to the nearest tenth.
A plane flies at 360 mph on a bearing of 240∘. The wind blows 60 mph toward the northeast. Find the ground speed and the bearing, to the nearest tenth.
A plane flies at 530 mph on a bearing of 200∘. The wind blows 70 mph toward the east. Find the ground speed and the bearing, to the nearest tenth.
A plane flies at 510 mph on a bearing of 290∘. The wind blows 70 mph toward the southeast. Find the ground speed and the bearing, to the nearest tenth.
A plane flies at 560 mph on a bearing of 290∘. The wind blows 30 mph toward the west. Find the ground speed and the bearing, to the nearest tenth.
A plane flies at 530 mph on a bearing of 100∘. The wind blows 70 mph toward the southeast. Find the ground speed and the bearing, to the nearest tenth.
A plane flies at 460 mph on a bearing of 040∘. The wind blows 60 mph toward the northeast. Find the ground speed and the bearing, to the nearest tenth.
A plane flies at 520 mph on a bearing of 080∘. The wind blows 80 mph toward the north. Find the ground speed and the bearing, to the nearest tenth.
A plane flies at 410 mph on a bearing of 100∘. The wind blows 30 mph toward the west. Find the ground speed and the bearing, to the nearest tenth.
The angle between a 10 lb force and a 60 lb force on a hook is 150∘. Find the magnitude of their resultant and the angle it makes with the 10 lb force.51.6 lb,144.4∘
Two ropes tied to a post pull with 50 lb and 55 lb, and the angle between the ropes is 45∘. Find the size of the resultant and its angle from the 50 lb rope.97.0 lb,23.6∘
Forces of 80 N and 35 N act on an object with an angle of 45∘ between them. Find the size of the resultant and its angle from the 80 N force.107.6 N,13.3∘
Two people push a cart, one with 10 lb and one with 85 lb, their pushes 100∘ apart. Find the size of the resultant and its angle from the 10 lb push.83.8 lb,93.3∘
Two ropes tied to a post pull with 75 N and 40 N, and the angle between the ropes is 80∘. Find the size of the resultant and its angle from the 75 N rope.90.9 N,25.7∘
Two people push a cart, one with 35 N and one with 50 N, their pushes 135∘ apart. Find the size of the resultant and its angle from the 35 N push.35.4 N,90.6∘
The angle between a 95 N force and a 50 N force on a hook is 80∘. Find the magnitude of their resultant and the angle it makes with the 95 N force.114.8 N,25.4∘
Forces of 80 lb and 15 lb act on an object with an angle of 150∘ between them. Find the size of the resultant and its angle from the 80 lb force.67.4 lb,6.4∘
The angle between a 45 N force and an 85 N force on a hook is 50∘. Find the magnitude of their resultant and the angle it makes with the 45 N force.119.0 N,33.2∘
Forces of 35 lb and 70 lb act on an object with an angle of 135∘ between them. Find the size of the resultant and its angle from the 35 lb force.51.6 lb,106.3∘
Find ground speed and bearing with wind
A plane flies at 310 mph on a bearing of 220∘. The wind blows 70 mph toward the south. Find the ground speed and the bearing, to the nearest tenth.about 366.4 mph on a bearing of 212.9∘
A plane flies at 300 mph on a bearing of 060∘. The wind blows 50 mph toward the north. Find the ground speed and the bearing, to the nearest tenth.about 327.9 mph on a bearing of 052.4∘
A plane flies at 360 mph on a bearing of 240∘. The wind blows 60 mph toward the northeast. Find the ground speed and the bearing, to the nearest tenth.about 302.4 mph on a bearing of 242.9∘
A plane flies at 530 mph on a bearing of 200∘. The wind blows 70 mph toward the east. Find the ground speed and the bearing, to the nearest tenth.about 510.3 mph on a bearing of 192.6∘
A plane flies at 510 mph on a bearing of 290∘. The wind blows 70 mph toward the southeast. Find the ground speed and the bearing, to the nearest tenth.about 447.5 mph on a bearing of 286.2∘
A plane flies at 560 mph on a bearing of 290∘. The wind blows 30 mph toward the west. Find the ground speed and the bearing, to the nearest tenth.about 588.3 mph on a bearing of 289.0∘
A plane flies at 530 mph on a bearing of 100∘. The wind blows 70 mph toward the southeast. Find the ground speed and the bearing, to the nearest tenth.about 588.7 mph on a bearing of 103.9∘
A plane flies at 460 mph on a bearing of 040∘. The wind blows 60 mph toward the northeast. Find the ground speed and the bearing, to the nearest tenth.about 519.8 mph on a bearing of 040.6∘
A plane flies at 520 mph on a bearing of 080∘. The wind blows 80 mph toward the north. Find the ground speed and the bearing, to the nearest tenth.about 539.7 mph on a bearing of 071.6∘
A plane flies at 410 mph on a bearing of 100∘. The wind blows 30 mph toward the west. Find the ground speed and the bearing, to the nearest tenth.about 380.5 mph on a bearing of 100.8∘
Forces of 30 N and 40 N act at right angles. Find the resultant and its angle from the 30 N force.
Put the 30 N force along the x-axis and the 40 N force along the y-axis, so the resultant is (30, 40).
Its size is the square root of 30 squared plus 40 squared, which is 50.
Its angle from the 30 N force is the inverse tangent of 40 over 30, about 53.1 degrees.
The answer is 50 N,53.1∘.
Where students go wrong
Adding the sizes. Two forces only add straight up when they point the same way; at any other angle the resultant is smaller. The other slip is measuring the resultant's angle from the wrong force, or from the wrong side of the x-axis when the direction angle is past 90 degrees.
Also called resultant of two forces, net force, adding forces or vector word problems.
What you can put on this worksheet
The resultant of two forces
Two ropes pull a crate at right angles to each other, one with 35 lb and the other with 120 lb. Find the size of the resultant and its angle from the 35 lb rope.→ 125 lb,73.7∘
Navigation with wind
A plane flies at 410 mph on a bearing of 200∘. The wind blows 70 mph toward the north. Find the ground speed and the bearing, to the nearest tenth.→ about 345.1 mph on a bearing of 204.0∘
Questions about these worksheets
Yes. Every resultant force sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.
Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.
Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.
Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.
Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.
That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the resultant force questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.
Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.
You pick from 2: the resultant of two forces and navigation with wind. Tick as many as you want and set how many of each, or let it spread them evenly.
Adding the sizes. Two forces only add straight up when they point the same way; at any other angle the resultant is smaller. The other slip is measuring the resultant's angle from the wrong force, or from the wrong side of the x-axis when the direction angle is past 90 degrees.