AP Physics 1/Kinematics/Vectors and Scalars

Educational level: this is a secondary education resource. In this lesson we will learn about vectors and scalars. Gaining a good understanding of these is crucial to physics and beyond. Below you will find a series of example problems, which you should always try to do first. Then you can read the solutions, followed by the contents. Finally, do the exercises at the end.

Example A Speed, such as "40 kph" or "3 m/s" is a scalar quantity. Velocity, such as "40 kph west" or " 3 m/s south" is a vector quantity. Compare these two values, then answer the question: What is a vector quantity? What is a scalar quantity? Example B Vector quantities are often represented by an arrow. These arrows are called vectors. The direction the arrow points represent the direction, and the length represents the magnitude. Consider the following: A plane is taxiing to the runway at 50 kilometers per hour due north. There is a crosswind with velocity 50 kilometers per hour east. What is the velocity of the plane? We know the answer to be about 70.71 kilometers due northeast. To get to this answer, we have to "add" the two vectors. Figure out how to do this. Example C A plane is flying at 900 kilometers per hour due north. There is s crosswind with velocity 50 kilometers per hour east. What is the velocity of the plane?

Alright. By now you should have tried each of these examples and you are ready to move on to reading the solutions. Let's see what you can learn from the content. Example A Speed, such as "40 kph" or "3 m/s" is a scalar quantity. Velocity, such as "40 kph west" or " 3 m/s south" is a vector quantity. Compare these two values, then answer the question: What is a vector quantity? What is a scalar quantity? Solution All we have to do to get to the solution is to compare and contrast them. What is different? We notice that velocity includes words like "West" and "South" - direction! It seems that scalar quantities only have magnitude, while vector quantities have both magnitude and direction.◊ In that example, we learned what scalar and vector quantities are. Of the two, vector quantities are the most important. Next up, we learn how to add vectors together. Example B Vector quantities are often represented by an arrow. These arrows are called vectors. The direction the arrow points represent the direction, and the length represents the magnitude. Consider the following: A plane is taxiing to the runway at 50 kilometers per hour due north. There is a crosswind with velocity 50 kilometers per hour east. What is the velocity of the plane? We know the answer to be about 70.71 kilometers due northeast. To get to this answer, we have to "add" the two vectors. Figure out how to do this. Solution We need to draw some vectors! When we do so, we get the following image to the right. We need to figure out how to "add" two vectors. After some trial and error, as well as exploration, you should have come up with one of the following methods for adding vectors together: Parallelogram Rule This rule states that when the two vectors are arranged like to the right, and when you form a parallelogram with the sides, the diagonal is the resultant. The resultant is just another word for the sum after you do vector addition. In the problem, we align the two vectors like so, then draw the diagonal to obtain our answer. Tip-to-Tail Method This method is one of the more commonly used ones. The method says that you must align the vectors tip-to-tail, like so. The line connecting the two endpoints, shown to the right, is the resultant. In the problem, we position the two vectors and then draw the hypotenuse to obtain our answer.◊ In a way, they are the same thing! How? The final example problem puts our already learnt content to the test. We will illustrate with the parallelogram rule. Example C A plane is flying at 900 kilometers per hour due north. There is a crosswind with velocity 50 kilometers per hour east. What is the speed of the plane? Solution We proceed as usual in this problem. The plane is flying due north at 900 kph, so we draw an arrow pointing up. The problem says that there is a crosswind blowing at 50 kph east, so we draw that like so. Finally, we construct our parallelogram and draw the diagonals to get to our answer, which is about 901.39 kilometers per hour.◊

Problems Do each of these problems on a separate piece of paper. Check your answers with the Solution Manual. 1. What is the resultant of the following vectors: 10 km/h north, 10 km/h east? 2. What is the resultant of the following vectors: 20 km/h west, 20 km/h south? 3. What is the resultant of the following vectors: 10 km/h heading 0, 20 km/h heading 90? 4. What is the resultant of the following vectors: 5 m/s heading 15, 10 m/s heading 180? 5. What is the resultant of the following vectors: 20 m/s heading 0, 5 m/s 250? 6. A force is acting on a pair of ropes as shown. Which rope has the greater tension? 7. A force is acting on a pair of ropes as shown. Which rope has the greater tension? 8. A force is acting on a pair of ropes as shown. Which rope has the greater tension? 9. A boat is in the ocean. It is being propelled by one motor at 25 kph north. A wind is acting on the boat at an angle of 60 degrees, shown below. The wind speed is 50 kph. What is the velocity of the boat?

Challenge Problems These problems are especially challenging! Hints are listed in the hints manual and are randomly numbered. 10. The Equilibrium Rule states that if an object is in equilibrium, then the net force is zero. Use this to answer this problem: George is hanging on a pair of ropes as shown. One of the ropes is half the length of the other and they form a right angle. George's body is 300 N. Calculate the force on each of the ropes. Hints: 50, 101

Go next Solution Manual: AP Physics 1/Solution Manual Hints Manual: AP Physics 1/Hints Manual Next Lesson: Lesson 2: Representing Motion Part 1

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