A race car accelerates uniformly from 13.5 m/s to 56.1 m/s in 3.92 seconds. Determine the acceleration of the car and the distance traveled.
When you see a problem with "uniform acceleration" you know you can use the kinematics equations. Those equations are as follows: 1) d = Vi*t + 1/2*a*t^2 2) d = (Vi+Vf)/2 * t 3) Vf = Vi + a*t 4) Vf^2 = Vi^2 + 2*a*d where d = distance; Vi = initial velocity; Vf = final velocity; a = acceleration; and t = time. For this problem, we will use equation number 3 to solve for the acceleration, since we have all of the other missing variables already. Plugging our numbers into equation number 3 gives us: 56.1 = 13.5 + a*3.92 Minus 13.5 from both sides 42.6 = a*3.92 Divide both sides by 3.92 a = 10.867 m/s^2 Now that we have the acceleration, we can use another equation to solve for the distance that the car traveled. Equation number 1 is the most commonly used kinematic equation, and will help us solve for the distance. Plugging in our numbers gives us: d = 13.5*3.92+(1/2)*10.867*3.92^2 Performing the squared: d = 13.5*3.92+(1/2)*10.867*15.366 Performing the multiplication: d = 52.92 + 89.493 d = 142.413 meters Answer: Acceleration = 10.867 m/s^2; Distance Traveled = 142.413 meters
What volume of 10.0 M H2SO4 is required to prepare 4.0 L of 0.50 M H2SO4? A) 0.20 L B) 0.40 L C) 0.50 L D) 1.0 L E) 4.0 L
The easiest way to deal with questions involving molarity is to find how many moles are on each side of the equation. We will start with the 4.0 L of 0.50 M solution. 0.50 M just means that there are 0.5 moles in each liter of solutin. We have 4.0 liters of this solution so 0.50 multiplied by 4.0 gives us 2.0 moles of solvent (H2SO4). Now we know that in our final solution we will need 2.0 moles of solvent. So how many liters of 10.0 M will give us 2.0 moles? 10.0 M means there are 10 moles in each liter. We need one fifth of that. So 1/5 of a liter, or 0.20 L. Answer: A) 0.20 L
For what values of "x" is the following equation true? 5(-3x - 2) - (x - 3) = -4(4x + 5) + 13
The equation we are given is: 5(-3x - 2) - (x - 3) = -4(4x + 5) + 13 Our first goal is to break everything down into simple terms without parantheses so that we can move terms around easier. To do this, we will use a technique called "factoring" "Factoring" means you multiply each term inside the parantheses by the one that is next to them. Fist we will factor "5(-3x - 2)", then "-(x-3)" and finally "-4(4x + 5)" 1) 5 multiplied by -3x gives us -15x, and 5 multiplied by 2 gives us -10 -----> -15x - 10. 2) The second term implies there is a -1 in front of (x-3) so -1 multiplied by x gives us -x and -1 multiplied by -3 give us +3 -----> -x+3. 3) -4 multiplied by 4x gives us -16x and -4 multiplied by 5 gives us -20 -----> -16x - 20. Now we rewrite the original equation with these new pieces that have no parantheses. -15x - 10 - x + 3 = -16x - 20 + 13 The next step is to "combine like terms" this means that you add/subtract all the terms with x's on each side, and all the terms without x's on each side. Left Side: -15x - x = -16x and -10 + 3 = -7 -----> -16x - 7 Right Side: -16x = -16x and -20 + 13 = -7 -----> -16x - 7 Now we have a problem, both sides are exactly the same! If we move one side over to the other we are left with 0 = 0. This means that no matter what number you choose x to be, that it will be an answer for the equation. Answer: The equation is true for ALL values of x.