Vector Examples Physics 1 Prepared by Vince Zaccone

Vector Examples Physics 1 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

b = opposite side Right Triangles c= hy po ten use θ a = adjacent side Pythagorean Theorem a 2+b 2=c 2 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Vector Components θ Here is how you will use right triangles in this course. Often you will need to find components of a vector. Or, given the components, you will need to find magnitude and direction for a vector. This is the biggest stumbling block for students in the Physics 6 series. A few examples follow. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Vector Components Here we have a projectile that is launched upward at an angle of 54°, and at an initial speed of 40 m/s. We want to find the horizontal and vertical components of this initial velocity vector. This is the first step in finding out where and when the projectile will land, and how high it will go. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Vector Components Here we have a projectile that is launched upward at an angle of 54°, and at an initial speed of 40 m/s. We want to find the horizontal and vertical components of this initial velocity vector. This is the first step in finding out where and when the projectile will land, and how high it will go. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Vector Components 3. 5 km 4. 5 km In this example, a cyclist heads due East for 4. 5 km, then turns and travels North for 3. 5 km. We need to find the magnitude and direction of the overall displacement vector. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Vector Components dy=3. 5 km θ dx=4. 5 km In this example, a cyclist heads due East for 4. 5 km, then turns and travels North for 3. 5 km. We need to find the magnitude and direction of the overall displacement vector. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Vector Components Here a few examples for practice. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Vector Components Here a few examples for practice. same triangle Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Vector Components Here a few examples for practice. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

A sailor in a small sailboat encounters shifting winds. She sails 2. 00 km east, then 3. 50 km southeast, and then an additional distance in an unknown direction. Her final position is 5. 80 km directly east of her starting point. Find the magnitude of the third leg of the journey. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

A sailor in a small sailboat encounters shifting winds. She sails 2. 00 km east, then 3. 50 km southeast, and then an additional distance in an unknown direction. Her final position is 5. 80 km directly east of her starting point. Find the magnitude of the third leg of the journey. y x The first step is to set up a coordinate system. The most convenient thing to do is just call East the x-direction and North the y-direction, as shown. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

A sailor in a small sailboat encounters shifting winds. She sails 2. 00 km east, then 3. 50 km southeast, and then an additional distance in an unknown direction. Her final position is 5. 80 km directly east of her starting point. Find the magnitude of the third leg of the journey. y x Now we look at each leg of the journey and find the components: First leg: This one is easy. 2. 00 km in the x-direction; 0. 00 km in the y-direction We could label everything to keep it all organized: If the first leg is represented by the vector V 1, we can write the components separately. V 1 x = +2. 00 km (note: positive indicates East – negative would be West) V 1 y = 0. 00 km Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

A sailor in a small sailboat encounters shifting winds. She sails 2. 00 km east, then 3. 50 km southeast, and then an additional distance in an unknown direction. Her final position is 5. 80 km directly east of her starting point. Find the magnitude of the third leg of the journey. y x The 2 nd leg has components in both x- and y-directions, so we can find the components from our triangle rules: V 2 x = V 2 y = Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

A sailor in a small sailboat encounters shifting winds. She sails 2. 00 km east, then 3. 50 km southeast, and then an additional distance in an unknown direction. Her final position is 5. 80 km directly east of her starting point. Find the magnitude of the third leg of the journey. y x The 2 nd leg has components in both x- and y-directions, so we can find the components from our triangle rules: V 2 x = +3. 50*cos(45°) ≈ 2. 47 km (positive for East) V 2 y = -3. 50*sin(45°) ≈ -2. 47 km (negative for South) Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

A sailor in a small sailboat encounters shifting winds. She sails 2. 00 km east, then 3. 50 km southeast, and then an additional distance in an unknown direction. Her final position is 5. 80 km directly east of her starting point. Find the magnitude of the third leg of the journey. y x The 3 rd leg has unknown components, but we can write them as unknowns: V 3 x = ? V 3 y = ? Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

A sailor in a small sailboat encounters shifting winds. She sails 2. 00 km east, then 3. 50 km southeast, and then an additional distance in an unknown direction. Her final position is 5. 80 km directly east of her starting point. Find the magnitude of the third leg of the journey. y x The next step is the only one where you have to think about the problem. Since we know that the total journey from start to finish is 5. 80 km East, we know the components: Vtotal, x = Vtotal, y = Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

A sailor in a small sailboat encounters shifting winds. She sails 2. 00 km east, then 3. 50 km southeast, and then an additional distance in an unknown direction. Her final position is 5. 80 km directly east of her starting point. Find the magnitude of the third leg of the journey. y x The next step is the only one where you have to think about the problem. Since we know that the total journey from start to finish is 5. 80 km East, we know the components: Vtotal, x = +5. 80 km Vtotal, y = 0. 00 km Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

A sailor in a small sailboat encounters shifting winds. She sails 2. 00 km east, then 3. 50 km southeast, and then an additional distance in an unknown direction. Her final position is 5. 80 km directly east of her starting point. Find the magnitude of the third leg of the journey. y x Finally we can just add everything up and solve for our unknowns. We will have 2 equations – one for each direction: V 1 x + V 2 x + V 3 x = Vtotal, x V 1 y + V 2 y + V 3 y = Vtotal, y Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

A sailor in a small sailboat encounters shifting winds. She sails 2. 00 km east, then 3. 50 km southeast, and then an additional distance in an unknown direction. Her final position is 5. 80 km directly east of her starting point. Find the magnitude of the third leg of the journey. y x Finally we can just add everything up and solve for our unknowns. We will have 2 equations – one for each direction: V 1 x + V 2 x + V 3 x = Vtotal, x V 1 y + V 2 y + V 3 y = Vtotal, y 2. 00 + 2. 47 + V 3 x = 5. 80 0. 00 – 2. 47 + V 3 y = 0. 00 V 3 x = +1. 33 km V 3 y = +2. 47 km (positive is East) (positive is North) Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

A sailor in a small sailboat encounters shifting winds. She sails 2. 00 km east, then 3. 50 km southeast, and then an additional distance in an unknown direction. Her final position is 5. 80 km directly east of her starting point. Find the magnitude of the third leg of the journey. y x Now that we have the components, the last step is to combine them to get the magnitude and direction of the vector for the 3 rd leg. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

A sailor in a small sailboat encounters shifting winds. She sails 2. 00 km east, then 3. 50 km southeast, and then an additional distance in an unknown direction. Her final position is 5. 80 km directly east of her starting point. Find the magnitude of the third leg of the journey. y x Now that we have the components, the last step is to combine them to get the magnitude and direction of the vector for the 3 rd leg. Use the Pythagorean Theorem to find the magnitude: For extra fun, find the angle of the vector too! V 3 = 2. 81 km Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

A sailor in a small sailboat encounters shifting winds. She sails 2. 00 km east, then 3. 50 km southeast, and then an additional distance in an unknown direction. Her final position is 5. 80 km directly east of her starting point. Find the magnitude of the third leg of the journey. y x Now that we have the components, the last step is to combine them to get the magnitude and direction of the vector for the 3 rd leg. Use the Pythagorean Theorem to find the magnitude: Use tan(θ) = Vy/Vx to find the direction tan(θ) = 2. 47/1. 33 Θ = tan-1(1. 86) Θ = 61. 7° North of East V 3 = 2. 81 km (this angle is measured counterclockwise from the positive x-axis) Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB