Chapter 6 – Vectors

1) What are vectors and scalars?

Vectors (vector quantities) and scalars (scalar quantities) are ways to measure things. The difference is that vector quantities require magnitude AND direction to be described, while scalars need only magnitude. Scalars are probably more common, and include things like time, mass and volume. Vectors are used to describe things like motion, velocity, and force, where direction is important. When dealing with scalars, you can use ordinary math to add, subtract, multiply or divide them (the mass of 5 bags with 5 kg of sand each is 5 x 5 = 25 kg). WIth vectors, you have to take their direction into account before you can combine them!

2) How do you use vectors to represent forces

Representing force with vectors is easy. The length of the vector, when used with a scale, gives the magnitude of the force, while the orientation of the vector and the arrowhead show the direction.

The vector length and direction are generally labeled, otherwise you are just guessing!

3) How do you use vectors to represent velocity?

Representing velocity with vectors is pretty much the same as representing force. You have a scale and a direction, and use them to show speed and direction. The illustration below shows 3 vectors: one for the plane, one for the tailwind (a wind going in the same direction as the object), and the resultant vector obtained by combining the first two!

4) How do you add vectors?

Sometimes, adding vectors is easy. If they are going the same direction, like above, all you have to do is add them together: 100 km/hr plus 25 km/hr equals 125 km/hr. It is also easy to do if the vectors are going in opposite directions; if the wind were going in the opposite direction, it would be 100 km/hr - 25 km/hr = 75 km per hour.

But vectors are usually not going to be quite so cooperative (how often is the wind going to blow in your exact direction, or in the exact opposite direction), so you need another method to combine them. To combine two we use geometric addition, where both vectors are redrawn as dashed lines with the tails of new vectors touching the heads of

their original opposites, making a parallelogram. You then draw a new vector, called the resultant, from the origin to the intersection of the dashed vectors.

If you have more than 2 vectors, you use a similar approach, but it doesn’t form a nice parallelogram. To find the resultant, draw a new vector from the starting point to the end. The order that you add the vectors doesn’t matter!

Chapter 6 Sections 5-7 – More on Vectors

1) What are a vector’s components?

Since you can combine two vectors into a resultant vector, it makes sense that any single vector can be broken down into components. The process of creating component vectors is called resolution. Resolution is used to see how much force is (or must be) applied to an object to get it to move at an angle.

Generally, when resolving a vector, it is broken into horizontal and vertical components. Start at the vector’s tail, and draw in a horizontal line headed toward the head of the vector. Than do the same thing with a vertical line. Then estimate the horizontal and vertical lengths of the vector, by running a protractor along the lines, and marking where a right angle line hits the head of the vector. The final product should look like the diagram below. By using the same scale as the original vector, you can estimate the horizontal and vertical force components of the original vector.

2) Why can you safely hang from a rope vertically, but have that rope break if you hang from it horizontally?

When something is suspended from a rope, the force with which the rope pulls up (Newton’s 3rd law) is the same as the weight of the object that is being suspended. This is called equilibrium. But if you hang from a horizontal rope, the rope pulls up from you in two directions, and the force of each of those pulls must be larger, because they must resolve to create a resultant vector going upward.

For example, if a 10 N weight is hanging on a single spring scale, it registers a force of 10 N. If

that same weight hangs vertically from 2 scales, each registers 5 N of force, for a total of 10 N. But if the two scales are each 60° from vertical (creating a 120° between them), each scale registers 10 N (30°-60°-90° triangle relationship), for a total of 20 N of force on the scales. Moving the scales further apart increases the force on them even more - at 75.5° from vertical, the force is 20 N per scale - and with something attached horizontally, the angle between the ends is 180°, the largest possible, so the force on the rope would be the largest possible, and it will probably break.

3) How do vector components affect weight?

While the weight of an object stays constant as long as its mass and gravity remain constant, the

forces caused by weight can vary. If the object is falling, the force pulling down on it is equal to its weight, and apart from air resistance, there are no other forces in play. If the object is on a flat, horizontal surface, the only active force on the object is the force of its weight pulling it down, and the horizontal components are zero. But if the object is on any kind of a slope, the weight pulling down on the object resolves into a force parallel to the slope, which causes the object to move down the slope, and a force perpendicular to the slope, which pushes the object against the slope. The steeper the slope, the larger the parallel vector, and the faster the acceleration of the object, with the maximum acceleration (9.8 m/s2) when the object is falling.

Chapter 6 Sections 8-9 – Projectile Motion

1) What is a projectile?A projectile is any object that has been shot, fired, or thrown in such a manner that it is moving in two directions, rather than in one direction, like an object that was simply dropped. Like a vector, the path of a projectile has both horizontal and vertical components.

2) How is the movement of a projectile different from an object that is just falling?

The horizontal component of a projectile’s motion is just like that for a ball rolled along a flat surface - if air resistance and friction are ignored, it moves the same distance each second (constant velocity).The vertical component of a projectile’s motion is just like that for a ball that has been dropped - if air resistance is ignored, the distance it moves each second increases according to gravity (acceleration) or in the case of a projectile shot upward, the speed first decreases until it reaches its highest point (where the speed is zero), then starts to increase.

The two components of a projectile’s motion are independent of each other. If you shoot an arrow across level ground and drop an arrow from the same height at the same time, they hit the ground at the same time.

3) How fast is does a projectile that was shot upward hit the ground?

A projectile fired upward follows a curving path (a parabola) upward and then back down to the ground. It has an initial speed upward, and slows down due to gravity, until its speed reaches zero, when the projectile is at its highest point. Then it accelerates downward due to gravity, and since it

is covering the same distance down as it did up, ittakes the same amount of time, and accelerates the same amount as it decelerated, so the speed at the ground (or more precisely, at the height from which it was fired) is exactly the same as the projectile’s initial velocity.

That is why firing a gun into the air is so dangerous - the bullets speed when it reaches the ground is the same as the speed with which it left the gun!

4) When firing a projectile, what angle gives you the most horizontal distance?

As you change the angle at which a projectile is shot upward, you change the shape of the projectile’s path. The smaller the angle from horizontal, the shorter the time the projectile moves before it hits the ground, thus the greater the angle, the longer the time it takes to hit the ground, and the higher the projectile goes before falling back down. Maximum distance comes from a 45° angle, and anything more or less will have a shorter range.

None of this takes air resistance or wind into account, which is why mortars and artillery generally need to fire a few test shots to zero in on their targets. Generally, in real life, projectiles do not travel as far as they do in “physics world,” and the speeds and ranges you calculate using formulas for speed and distance and are maximums rather than averages.