A quadratic function has solutions at x = -1 and x = 3. The function is not stretched or squeezed. Which two points could be the vertex of this parabola?
The thing I love about quadratic functions, which look like parabolas on a graph, is that they are vertically symmetrical. This makes them both pleasing to look at and also predictable. I see that this quadratic function has "solutions" at x = -1 and x = 3. That's another way of saying that -1 and 3 are the x values where the parabola crosses the y axis on a graph. Using this information, I can find the x-value of location where the graph has either its highest or lowest point, also known as the vertex. Because the parabola has to be symmetrical, the vertex must be at x = 1, which is right in the middle of x = -1 and x = 3. The y value is a bit trickier to figure out because we don't have the equation of the quadratic function. Let's think about what the parabola would look like for the function f(x) = x^2. The vertex is at (0,0). At x = 1, y = 1. at x = 2, y = 4. In our parabola, x = -1 and x = 3 are each two away from x = 1. This means that the vertex will have a y-value 4 units below these points or, if the x^2 term of our quadratic function is negative, 4 units above. The means the vertex could be at (1, -4) or (1,4).
Your teacher holds a block and a ball, each with the same mass, at the top of a ramp. When she lets go, the block slides down the ramp without friction and the ball rolls down the ramp without slipping. When they reach the bottom, are the objects traveling at the same speed or a different speed? If they travel at different speeds at the bottom of the ramp, which is faster?
Often, the hardest part of any physics problem is deciding what kind of problem it is. In introductory physics courses (even advanced ones) problems typically fall into one of three categories: force problems, momentum problems, or energy problems. There aren't forces or masses given in this problem. It would be challenging to analyze this from a forces perspective. Often thought not always, momentum problems involve objects coming into contact with one another or interacting in some way. The block and the ball aren't interacting with one other, so I'm not sure I would use a momentum perspective to think through this problem either. Let's analyze the situation from an energy perspective, starting with what we know about energy. We know that, when no external forces act on a system, the total energy of a system does not change (Law of Conservation of Energy) but the forms of energy that make up that total can change. This means 2 things. First, we should "define our system" so there are no external forces. Second, we should find out what kinds of energy there are in that system before the block and ball start moving, and at the bottom of the ramp. "Defining the system" means we choose which objects are in our system. We will define the block's system as the block and Earth, which keeps the force due to gravity internal to the system. Likewise, we will define the ball's system as the ball and Earth. There are two main kinds of energy that could be a part of the system: potential energy and kinetic energy. Kinetic energy is the energy an object has due to its motion. At the top of the ramp, the ball and block do not have kinetic energy because they are not moving. They do, however, have gravitational potential energy (GPE) because Earth is part of the system. GPE is proportional to mass and height. Since the block and the ball both have the same height and mass, they have the same GPE. As the block slides down the ramp and as the ball rolls down it, the objects' GPEs decrease because their heights decrease. Because both reduce height by the same amount, the GPEs decrease by the same amount. We know the total energy in the block-Earth and ball-Earth system has to stay the same, which means the GPE transforms into another form of energy. In this case, it's kinetic energy (KE), or energy due to an object's motion. Generally speaking, the faster an object goes, the greater its KE. It is fair to assume that the GPE of the ball and block transformed into KE and therefore conclude that the ball and block have the same amount of KE at the bottom of the ramp. You might get tricked into assuming this means that the ball and block have the same velocity at the bottom of the ramp. Unfortunately, it's not that easy. There are two primary types of KE: linear and rotational. Linear KE is related to linear velocity, or the speed at which an object goes from one location to another. Rotational KE is related to rotational velocity, or the velocity of an object as it spins. The block does not have rotational KE but the ball does. This means that all the GPE of the block converted into linear KE, but the GPE of the ball converted into linear KE and rotational KE. Even if the ball has only a small amount of rotational KE, it will have less linear KE than the block, meaning it goes slower. This is a very long-winded way of saying the block and ball have different speeds at the bottom of the ramp, and the block is traveling faster than the ball.
You and your friend are hanging out at the beach when she challenges you to a race to see who can get to the ice cream stand first. The ice cream stand is 130 feet away to the northeast of your current location. Your friend runs straight for the ice cream stand at 5 feet per second. You can run about as fast as her but she got a 3 second head start so you can't catch her by running straight to the stand like her. Instead, you run 50 feet directly north until you reach the boardwalk where you turn east and run faster, about 10 feet per second. Who wins the race?
Before we start to solve this problem, let's start by finding out all the information we can. I always like to draw a picture of the situation. You know that your friend runs northeast 130 feet and you run 50 feet directly north to start, with both of you starting in the same place. You know you run directly east after getting to the boardwalk, but you don't know how far your position is to the ice cream stand. That's not a problem! If you draw the picture of you and your friend's paths, you'll notice that you have a right triangle. The hypotenuse is 130 ft. and one of the sides is 50 feet. If you have memorized the special right triangles, you'll know that the remaining side is 120 feet. If you haven't memorized the special right triangles, you can use the Pythagorean Theorem (a^2 + b^2 = c^2) to find the remaining side. Now we know: Your friend's distance and speed: 130 ft at 5 ft/s Your distance and speed: 50 ft at 5 ft/s, then 120 ft at 10 ft/s You can solve this problem completing calculations, with equations, and even by creating a graph. Let's solve with calculations. Every second, your friend travels 5 feet. It will take her 130 ft / 5 ft per second = 26 seconds to reach the ice cream stand. Not sure how I got this? Try a different way! Each 5 feet takes your friend 1 second. After 1 second, she travels 5 feet. After 2 seconds, she travels 10 feet. Keep going until you get to 130 feet. Either way, you get 26 seconds. Your friend takes 26 seconds but got a 3 second head start. That means you only have 26 s - 3 s = 23 seconds to get to the ice cream stand. Can you make it? Let's see! We will break your journey into 2 parts: the part when you're traveling north on the sand and the part when you're traveling east on the boardwalk. On the sand, you travel 50 feet at a rate of 5 feet every second. Using the same method as a above, we find that you will reach the boardwalk in 50 ft / 5 ft per second = 10 seconds. On the boardwalk, you travel 120 feet at a rate of 10 feet per second. This takes you 120 ft / 10 ft per second = 12 seconds. Combined, your journey takes 10 seconds + 12 seconds = 22 seconds, which means you beat your friend by 1 second. Congrats!