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.
Forces of 10 lb and 60 lb act on an object with an angle of 150∘ between them. Find the size of the resultant and its angle from the 10 lb force.
A plane flies at 590 mph on a bearing of 100∘. The wind blows 60 mph toward the northeast. Find the ground speed and the bearing, to the nearest tenth.
Forces of 100 N and 50 N act on an object with an angle of 80∘ between them. Find the size of the resultant and its angle from the 100 N force.
A plane flies at 250 mph on a bearing of 150∘. The wind blows 30 mph toward the north. Find the ground speed and the bearing, to the nearest tenth.
Forces of 15 N and 45 N act on an object with an angle of 130∘ between them. Find the size of the resultant and its angle from the 15 N force.
A plane flies at 310 mph on a bearing of 070∘. The wind blows 20 mph toward the south. Find the ground speed and the bearing, to the nearest tenth.
Forces of 85 lb and 35 lb act on an object with an angle of 50∘ between them. Find the size of the resultant and its angle from the 85 lb force.
A plane flies at 460 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.
Forces of 50 lb and 80 lb act on an object with an angle of 150∘ between them. Find the size of the resultant and its angle from the 50 lb force.
A plane flies at 290 mph on a bearing of 270∘. The wind blows 30 mph toward the east. Find the ground speed and the bearing, to the nearest tenth.
Forces of 30 N and 20 N act on an object with an angle of 120∘ between them. Find the size of the resultant and its angle from the 30 N force.
A plane flies at 430 mph on a bearing of 050∘. The wind blows 30 mph toward the north. Find the ground speed and the bearing, to the nearest tenth.
Forces of 90 lb and 30 lb act on an object with an angle of 150∘ between them. Find the size of the resultant and its angle from the 90 lb force.
A plane flies at 490 mph on a bearing of 150∘. The wind blows 40 mph toward the northeast. Find the ground speed and the bearing, to the nearest tenth.
Forces of 95 lb and 75 lb act on an object with an angle of 140∘ between them. Find the size of the resultant and its angle from the 95 lb force.
A plane flies at 540 mph on a bearing of 330∘. The wind blows 20 mph toward the north. Find the ground speed and the bearing, to the nearest tenth.
Forces of 35 lb and 90 lb act on an object with an angle of 80∘ between them. Find the size of the resultant and its angle from the 35 lb force.
A plane flies at 580 mph on a bearing of 250∘. The wind blows 30 mph toward the northeast. Find the ground speed and the bearing, to the nearest tenth.
Forces of 20 lb and 100 lb act on an object with an angle of 45∘ between them. Find the size of the resultant and its angle from the 20 lb force.
A plane flies at 350 mph on a bearing of 050∘. The wind blows 40 mph toward the east. Find the ground speed and the bearing, to the nearest tenth.
Forces of 10 lb and 60 lb act on an object with an angle of 150∘ between them. Find the size of the resultant and its angle from the 10 lb force.51.6 lb, 144.4∘
A plane flies at 590 mph on a bearing of 100∘. The wind blows 60 mph toward the northeast. Find the ground speed and the bearing, to the nearest tenth.about 626.3 mph on a bearing of 095.5∘
Forces of 100 N and 50 N act on an object with an angle of 80∘ between them. Find the size of the resultant and its angle from the 100 N force.119.3 N, 24.4∘
A plane flies at 250 mph on a bearing of 150∘. The wind blows 30 mph toward the north. Find the ground speed and the bearing, to the nearest tenth.about 224.5 mph on a bearing of 146.2∘
Forces of 15 N and 45 N act on an object with an angle of 130∘ between them. Find the size of the resultant and its angle from the 15 N force.37.2 N, 112.0∘
A plane flies at 310 mph on a bearing of 070∘. The wind blows 20 mph toward the south. Find the ground speed and the bearing, to the nearest tenth.about 303.7 mph on a bearing of 073.5∘
Forces of 85 lb and 35 lb act on an object with an angle of 50∘ between them. Find the size of the resultant and its angle from the 85 lb force.110.8 lb, 14.0∘
A plane flies at 460 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 441.0 mph on a bearing of 191.4∘
Forces of 50 lb and 80 lb act on an object with an angle of 150∘ between them. Find the size of the resultant and its angle from the 50 lb force.44.4 lb, 115.7∘
A plane flies at 290 mph on a bearing of 270∘. The wind blows 30 mph toward the east. Find the ground speed and the bearing, to the nearest tenth.about 260.0 mph on a bearing of 270.0∘
Forces of 30 N and 20 N act on an object with an angle of 120∘ between them. Find the size of the resultant and its angle from the 30 N force.26.5 N, 40.9∘
A plane flies at 430 mph on a bearing of 050∘. The wind blows 30 mph toward the north. Find the ground speed and the bearing, to the nearest tenth.about 449.9 mph on a bearing of 047.1∘
Forces of 90 lb and 30 lb act on an object with an angle of 150∘ between them. Find the size of the resultant and its angle from the 90 lb force.65.8 lb, 13.2∘
A plane flies at 490 mph on a bearing of 150∘. The wind blows 40 mph toward the northeast. Find the ground speed and the bearing, to the nearest tenth.about 481.2 mph on a bearing of 145.4∘
Forces of 95 lb and 75 lb act on an object with an angle of 140∘ between them. Find the size of the resultant and its angle from the 95 lb force.61.1 lb, 52.1∘
A plane flies at 540 mph on a bearing of 330∘. The wind blows 20 mph toward the north. Find the ground speed and the bearing, to the nearest tenth.about 557.4 mph on a bearing of 331.0∘
Forces of 35 lb and 90 lb act on an object with an angle of 80∘ between them. Find the size of the resultant and its angle from the 35 lb force.102.1 lb, 60.3∘
A plane flies at 580 mph on a bearing of 250∘. The wind blows 30 mph toward the northeast. Find the ground speed and the bearing, to the nearest tenth.about 553.0 mph on a bearing of 251.3∘
Forces of 20 lb and 100 lb act on an object with an angle of 45∘ between them. Find the size of the resultant and its angle from the 20 lb force.115.0 lb, 37.9∘
A plane flies at 350 mph on a bearing of 050∘. The wind blows 40 mph toward the east. Find the ground speed and the bearing, to the nearest tenth.about 381.5 mph on a bearing of 053.9∘
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 vector, resultant of two forces, net force, adding forces or vector word problems.
What you can put on this worksheet
The resultant of two forces
Forces of 35 lb and 120 lb act on an object at right angles. Find the size of the resultant and its angle from the 35 lb force.→ 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.