Please list newtons laws of motion and give a real world example of each.
Newtons first law is defined as an object in motion will stay in motion and an object at rest will stay at rest unless an outside force is acted upon it. This is also known as inertia. This can be demonstrated in the real world with a car accident. If a person or object is not attached to the car with a seat belt or strap, and the car gets stopped by a force like another car or wall, the car will stop but everything else not attached to the car will continue moving at the same speed the car was. Newtons 2nd law is the mathematical law that the net force is equal to the mass times the acceleration, F = ma. This would mean if a 10kg mass was accelerating at 10m/s^2, the mass would have a force of 100 Newtons. Also this helps explain that there is no net force if an object is moving at constant speed, meaning no acceleration. Newtons third law states that for every action there is an equal and opposite reaction and that is easily shown in a Newtons cradle. Usually about 5 steel balls are hanging and if you lift one and let it fall, the energy travels and the other end will shoot a ball out with the equal reaction.
Factor and find the solutions to x^2 -10x + 21 and explain the connection between a polynomials factors and zeros.
In order to answer this question you need to know how to factor trinomials (an algebraic expression with 3 terms) to factor you need to see what expressions multiply to give you back the trinomial. To factor you need to figure out what multiplies to be the 3rd term 21, and adds to be the middle term -10. The factors of 21 are 21 and 1 or 3 and 7. However our middle number is negative so we could also consider -3 and -7 which does add to -10 and multiplies to 21. Put these into two binomials and you have the answer of (x-3)(x-7). Solutions, roots, and zeros are other words for x intercepts. Factors set equal to zero and solved for x will give you the x intercepts of a function. One could graph the trinomial and visually see the x intercepts or set (x-3) and (x -7) equal to zero and get x = 3 and x = 7 as the solutions.
A cannonball is shot horizontally at a speed of 137 m/s off of a 230 meter cliff. What are the horizontal and vertical components of the initial velocity and how far did the cannon hit the ground from the base of the cliff?
This question requires the knowledge of 2 dimensional motion, sometimes called projectile motion. Since the cannonball is shot horizontally and not at an angle, the components of the velocity are very simple, the horizontal velocity or Vx is 137 m/s, the vertical component, Vy, is zero. The Vx will stay 137 m/s while the vertical will gain a magnitude as the cannonball falls. To find how far the cannonball goes, also called the range, you need to know the hang time of the cannonball which can be found from manipulating the kinematics equations to t = sqrt(2H/g). g is the acceleration due to gravity since the cannonball is falling freely and is a constant on earth, it is 9.8 m/s^2. So by plugging in the height (230m) and g, the hang time is 6.85 seconds. Then to find the range, you simply multiply the Vx and the time since velocity has a relation as the distance divided by the time. The answer is R = 938.45 meters.