Name: ________________________________________________ Science Teacher: ____________________ How to Solve Formulas Using the Triangle Method Let’s look at the formula for speed: Distance (d) Speed (s) = Time (t) Now make a triangle with a T: Now put the formula in the triangle: Now, all you have to do is to put the numbers in and do the math. For example: John traveled 10 miles in 5 minutes. What was his speed? ? Cover up the one you don’t know (speed) and use the triangle to either divide or multiply: You divide Time into Distance 10 miles 5 minutes 2 miles/minute (top unit/bottom unit) What if you have the speed, but not the distance? Use the triangle: Liam drove 60 miles per hour (mph). He drove for 2 hours. How far did he go? ? ? Cover up the one you need (distance). What does the triangle say to do? Multiply! 60 mph x 2 hrs = 120 miles (the unit of distance you need) Try this: Bob rode his bike at a speed of 10 mph. He rode 5 miles. How long did it take him? What are you going to do? 5 miles ÷ 10 mph = .5 hours ? You can use the triangle with other formulas. For example, density. What is density? It is how close the particles are in the volume of an object. For example, let’s take the air in a basketball. If you put the same amount of air in a baseball, the particles of air would be closer together. So the baseball is denser than a basketball, because it has more particles in the space available. Because volume is part of density, then that means we are “filling up” a container. If we are filling the container with liquid, then we may use liters or milliliters (think about Coca Cola containers.) Because a liquid fills up a container automatically, we don’t have to put cubed (3) next to the liters. It is already known. If we are filling up the container with a solid, then we usually use meters are centimeters cubed (because you are filling up the bottom and the sides.) So the ending unit would be m3 or cm3. Sometimes, we are filling up a container with a solid or a liquid, we say cc (for centimeters cubed.) A good example of this is when you get a shot of medicine. The formula for density is Mass (M) Density (d) = Volume (V) Can we use the triangle for this? Sure! Try this problem: The mass of object #1 is 40g. The volume is 2 cm3. What is the density of this object? Plug it in: Density = 40 g 2 cm3 = 20 g/cm3 ( top unit/ bottom unit) Try the triangle for these problems. How would you solve them? ? 1) The density of an object is 50 g/mL. The mass of the object is 2 g. What is the volume? Show me how you solve it: ? Answer: _________________________ (don’t forget the ending unit!) 2) The density of the object is 4 mg/cc. The volume of the object is 2 cc. What is the mass? Show me how you solve it: ? Answer: _________________________ (don’t forget the ending unit!) Let’s try something different, such as force. Force = mass x acceleration. Since we know that mass and acceleration have to be next each other to be multiplied, you can put it in the triangle like this: Now you can solve any force problem. 1) Sam’s mass is 50 kg, and he is acceleration is 5m/s2. What is his force? Answer: _________________________ (don’t forget the ending units!) 2) The bowling ball has a force of 30 kg/ft/sec2. It is rolling at 5 ft/sec2. What is its mass? ? Answer: _________________________ (don’t forget the ending units!) Acceleration is a change in speed. Think about all the times you change speed. You can speed up, or slow down, or even turn (change direction) will change your speed. So you can accelerate when you 1) Speed up (positive acceleration) 2) Slow down (negative acceleration or deceleration) 3) Change direction (whenever you turn, even going in a circle.) In order to figure out the rate of acceleration, you have to know your ending speed first. Then you look at the speed you started with, and finally, how long it took you to change your speed (time). So the formula for acceleration is Final velocity – starting velocity Time Yes, you can use the triangle: Because you have time on both the top (difference in speed over time), you will have the time twice as an ending unit. If the ending unit is the same, then you can just square (2) it. Example: 60 meters/sec 2 secs = 30 m/s/s or 30 m/s2 This means during each second of time, the speed increased by 30 meters per second. Vf - Vi A T Use the triangle to solve these acceleration problems: 1. Sherri began driving her car at a speed of 30 m/h. After 5 minutes, ? She was driving at a speed of 40 m/h. What was her rate of acceleration? Use the triangle: 40 mph – 30 m/h 10 m/h 5 minutes = 5 min = 2 m/h/m (how come the time wasn’t squared?) 2. The truck was traveling at 50 m/h when it had to apply its breaks. Within 2 minutes it came to a full stop. What was its rate of acceleration? (what is the ending speed?) ? 0 m/h - 50 m/h 2 min Can you subtract 50 from a 0? Sure – make it a negative number? -50 m/h 2 min = _________________ So, what does it mean when you get a negative number? Yes, You are slowing down (negative acceleration or deceleration) You will show that you really understand the triangle method if you can solve this problem: 3. Bob ran for 20 minutes. His rate of acceleration was 5m/min/min (or 5m/min 2). What was the difference in Bob’s velocity? Use the triangle to solve this problem: Answer: ______________________________ (don’t forget the ending unit) See how easy it is to use the triangle method! ?