Chapter 3 Two-Dimensional Motion and Vectors

Section 3-1: Introduction to Vectors Physical quantities such as length, area, volume, mass, density, and time can be expressed in terms of magnitude alone as single numbers with suitable units. Ex. The length of a table can be completely described as 1.5 m. Scalar quantities-- quantities that can be expressed completely by single numbers with appropriate units.

Scalar is derived from the Latin word scala which means “ladder” or “steps” and implies magnitude. Scalars: mass of steel block is 54 kg, the temperature is 50 degrees Celsius. Other physical quantities such as force, velocity, acceleration, electric field strength, and magnetic induction cannot be completely described in terms of magnitude alone. In addition to magnitude these quantities always have a specific direction.

Vector quantities– quantities that require magnitude AND direction for their complete description. A vector is usually shown as an arrow. The length of the arrow represents the magnitude of the vector, while the orientation of the arrow shows direction. North is usually toward the top of the page, east is toward the right, south is toward the bottom, and west is toward the left.

When 2 or more vectors act at the same point, it is possible to find a single vector that produces the same effect as the combination of separate vectors. Each separate vector is called a component. The single vector that produces the same result as the combined components is called the resultant.

When 2 or more vectors act at the same point and in the same direction, the magnitude of the resultant is the algebraic sum of the magnitudes of the components. Ex. Component A acts 8 units northward, component B acts 3 units northward.

When 2 or more vectors act at the same point but in opposite directions, the magnitude of the resultant is the algebraic difference of the component magnitudes. With the resultant having the direction of the larger component. To subtract a vector add its opposite. Ex. Component A acts 8 units northward and component B acts 3 units southward.

When 2 components do not act along a straight line, the resultant cannot be found through simple addition and subtraction. We must use the parallelogram method or the head to tail method to add the vectors. Ex. Component A acts 8 units northward and component B acts 3 units eastward. With the parallelogram method, the diagonal of the parallelogram is the resultant.

The magnitude and direction of the resultant can be found in 2 ways. Graphic solution– if the vector diagram is drawn accurately to scale, the resultant can be measured directly with a ruler and a protractor. Trigonometric solution– using trig ratios to calculate the magnitude and direction of the resultant.

The parallelogram method can also be used when vectors act at angles other than right angles.

To find the resultant of two or more vectors that act at the same point, it is very helpful to place them head to tail.

With the head to tail method, it does not matter in what sequence the arrows are drawn, as long as the direction of each arrow is maintained when it is transposed. The resultant is the arrow that is made by starting at the tail of the first component and drawing a straight line to the head of the last component.

Resolving Vectors into Components Any vector can be completely described by a set of perpendicular components. When a vector points along a single axis, the second component of the vector is equal to zero. By breaking a single vector into 2 components, or resolving it into its components, an object’s motion can sometimes be described more conveniently in terms of direction, such as north to south or east to west.

To illustrate this point, let’s examine a scene on the set of an action movie. For this scene, a biplane travels at 95 km/h at an angle of 20° relative to the ground. Attempting to film the plane from below, a camera team travels in a truck, keeping the truck beneath the plane at all times. How fast must the truck travel to remain directly below the plane? To find out the velocity that the truck must maintain to stay beneath the plane, we must know the horizontal component of the plane’s velocity.

The key to solving this problem is to recognize that a right triangle can be drawn using the plane’s velocity and its x and y components. The situation can then be analyzed using trig.

Adding Vectors That Are Not Perpendicular Until this point, the vector-addition problems concerned vectors that are perpendicular to one another. However, many objects move in one direction, and then turn at an acute angle before continuing their motion. Because the original vectors do not form a right triangle, it is not possible to directly apply the sine, cosine, and tangent functions when adding the original two vectors.

Determining the magnitude and the direction of the resultant can be achieved by resolving each of the vectors into their x and y components. Then, the components along each axis can be added together. These vector sums will be the two perpendicular components of the resultant. The magnitude and direction of the resultant can be found using the trig functions.

Section 3-3: Projectile Motion In this section, we will focus on the form of two-dimensional motion called projectile motion. Objects that are thrown or launched into the air and are subject to gravity are called projectiles. Projectile motion– free fall with an initial horizontal velocity.

The path of a projectile is a curve called a parabola. Many people mistakenly believe that projectiles eventually fall straight down. However, if an object has an initial horizontal velocity in any given time interval, there will be horizontal motion throughout the flight of the projectile. For the purposes of samples and exercises in this book, the horizontal velocity of the projectile will be considered constant.

The horizontal and vertical motions are completely independent of each other.

This velocity would not be constant if we accounted for air resistance. Because of air resistance, the true path of a projectile traveling through earth’s atmosphere is not a parabola.

If you drop a bullet from shoulder height and shoot a bullet from shoulder height, which will hit the ground first? All objects released from the same height, regardless of their horizontal velocity, will reach the ground at the same time. All objects have the same acceleration due to gravity, m/s 2.

The constant acceleration equations from chapter 2 can be used to solve for the vertical motion and horizontal motion of a projectile separately.

Suppose a rifle bullet is fired horizontally with a velocity of 1250 m/s. Neglecting air resistance, the bullet travels 1250 m horizontally by the end of the first second. Immediately after it leaves the muzzle of the gun, the force of gravity begins to accelerate the bullet toward the earth. This force is vertical. During the first second, the bullet drops 4.90 m

In two seconds, the bullet travels 2500 m horizontally, at which time it also drops vertically through a distance of 19.6 m. Over short distances the path of a high- velocity projectile approximates a straight line, but over greater distances, its path is noticeably curved. If you are firing a gun at a target that is some distance away, what must you do?

If the line of sight to a target is horizontal, a projectile must be fired at a small upward angle in order to compensate for the downward acceleration due to the force of gravity of the projectile. If the same size ammunition is used, the size of this angle will depend on what? The size of the angle depends on the distance to the target.

The muzzle velocity, v m, is resolved into the horizontal component, v x, and the vertical component, v y. The range is the horizontal displacement. The path of the projectile is called the trajectory.

We can use components to analyze objects launched at an angle. It can occur that a projectile is launched at an angle to the horizontal. When launched at an angle to the horizontal, the projectile has an initial vertical component of velocity as well as a horizontal component of velocity. Suppose the initial velocity vector makes an angle θ with the horizontal.

To analyze the motion of such a projectile, the object’s motion must be resolved into its components. The sine and cosine functions can be used to find the horizontal and vertical components of the initial velocity. v x = v i cosθ and v y,i = v i sinθ These values for initial velocity can then be substituted into the previous equations.