Some formulas you'll often use in algebra or everyday calculations include the rate of speed of an object moving for a given time and distance. These concepts are probably familiar, particularly if you're a fan of speed!\u00a0We'll walk you through\u00a0determining the rate of speed formulas in algebraic calculus.<\/p>\n
The difference between two identical objects that are moving at the same time is the distance they cover. You already know that the object moving faster will go\u00a0longer distances than the one moving slowly, if the time they're given is the same.<\/p>\n
If that doesn't make sense, you can think of it as the one moving faster, getting to its destination sooner than the slow one. So, when looking at speed, you'll need to\u00a0involve the\u00a0distance traveled and time taken<\/a>.<\/p>\n In a given span of time, double the speed translates to doubling the distance traveled. This also means that the time taken is halved, meaning your object is now twice as fast.<\/p>\n Speed is related to velocity, and that\u2019s why the symbol v is used to represent it in symbolic calculations. When determining the rate of speed formulas, keep the following two rules in mind;<\/p>\n To give a symbolic form definition of speed, combine these two rules to derive; v\u00a0=\u00a0s\/t<\/p>\n Though this is not the final definition, you can also define speed as the rate of change in distance within a specified time.<\/p>\n To calculate an object's speed includes knowing how far the distance traveled is and how long it's taken to cover that. Faster speed corresponds to further distance and sooner time.<\/p>\n Suppose a question is posed that you're driving a car from Boston to New York, a journey whose distance is known as 200 miles. Your trip ends up taking two hours, and you want to calculate the speed.<\/p>\n We'll use this formula:\u00a0\u00a0v<\/em>\u00a0=\u00a0s\/t<\/em><\/p>\n \u00a0\u2248\u00a0200\u00a0m\/4\u00a0hour\u00a0=\u00a050 m\/h<\/em><\/p>\n That's the equation's product, but is the speed of 50 miles per hour accurate for the entire 4-hour journey?<\/p>\n This speed can only be possible if you drove the entire way from Boston to New York using cruise control, and traffic moved out of the road for you. Since any hypothetical journey will have variations in speed, 50 miles per hour is your\u00a0average rate of speed<\/a>\u00a0for the trip.<\/p>\n You can modify the speed rate equation to further emphasize this:<\/p>\n \u00afv\u00a0=\u2206s\/\u00a0\u2206t<\/em><\/p>\n Look at the bar above the speed symbol (\u00afv), which indicates that you are calculating mean or average. The delta (\u2206) symbol represents the change made or distance covered.<\/p>\n When you look at the car's speedometer as you're driving, it shows the speed determined over a very short interval. This is called instantaneous speed, whose definition within your vehicle is limited to the measuring instrument's sensitivity.<\/p>\n Since an instant should be as close to zero as possible, it's easy to imagine the average speed of very tiny intervals. You can\u00a0effectively calculate instantaneous speed<\/a>\u00a0by writing it symbolically:<\/p>\n \u00a0v = lim\u00a0 \u00a0\u00a0\u00a0\u2206s\/\u2206t\u00a0 = ds\/dt<\/em><\/p>\n \u2206t\u21920\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0<\/em><\/p>\n In calculus, you can read this formula as speed being the first derivative of distance traveled with respect to the time taken.<\/p>\n When determining the rate of speed formulas in algebra, the most commonly used expression is as follows:<\/p>\n distance = rate x time.<\/p>\n As long as you divide the same non-zero elements of each side of an equation, it remains true. This formula can therefore be written in a variety of ways.<\/p>\n Rate = distance\u00a0\u00a0\/ time<\/p>\n Rate is, therefore, the distance, which can be in miles, kilometers, feet, etc. This is divided by the time taken in seconds, minutes, or hours.<\/p>\n The speed rate can be written as a fraction with a denominator of time and a numerator of distance units.\u00a0An example of this is when you say you've driven 25 miles in one hour. You can also convert the rate units when time is given in minutes and distance in feet.<\/p>\n Expressing the units must remain true when using the formula rate of speed x time = distance.\u00a0If a car were to travel at 30 miles per hour, you'd be able to figure out what distance is covered in say, 2 hours.<\/p>\n 30\u00a0 miles\/hour\u00a0 \u00a0x 2 hours = 60 miles<\/em><\/p>\n The hours will cancel out and leave only miles.<\/p>\n Rate is a type of ratio, and as such, is used in many everyday scenarios,\u00a0 including\u00a0grocery shopping, medicine, and traveling.\u00a0Many activities will involve some form of rate.\u00a0The heart rate is the number of heartbeats per second, while a babysitter will charge a specific rate which is a dollar amount per hour.<\/p>\n To determine the speed formulas rate, you'll use rates such as\u00a0feet per second or miles per hour<\/a>.\u00a0 You can find the clue that you're dealing with a rate when you come across the word 'per.'<\/p>\n In mathematical problems, the word per is replaced by a forward dash (\/).\u00a0When determining the rate of speed formulas, the rate is equal to the distance over time. The formula can be written like this:<\/p>\n r (rate of speed) = d (distance) \/ t (time)<\/p>\n Or you can use (d = rt) for solving distance and (t=d\/r) for time.\u00a0This is a formula derived from cross products and proportions, and one step of the multiplication has been handled for you.<\/p>\n If you rode a bike for 24 miles and spent 2 hours on the road, calculate your speed rate?<\/p>\n Using the r=d\/t formula, your speed rate becomes 24miles divided by the 2 hours, which equals 12 miles per hour.<\/p>\n r = 24 miles \/ 2 hours = 12 mph<\/em><\/p>\n What if you are given the speed rate, such as riding your bike for 4 hours at 10 miles per hour? You would use the d=rt formula this time:<\/p>\n d = 10 mph x 4 hours = 40 miles<\/em><\/p>\n If you were to travel at a speed of 12 miles per hour for 18 miles, how much time did you take?\u00a0Use the t=d\/r formula:<\/p>\n t = 18 miles \/ 12 mph = 1 \u00bd hours<\/em><\/p>\n There is always confusion when looking at velocity and speed, but these concepts differ slightly. While velocity takes account of the direction traveled by an object, speed doesn't. While speed is directly distance over time, velocity is the\u00a0change of an object's position over time<\/a>.<\/p>\n Let's say you drive from your home to your workplace and back again within an hour for the 20 miles distance.\u00a0Calculate your average speed.<\/p>\n Since you\u2019ve been given the total traveled distance and the time taken to cover it, all you need to do is to add the rate of speed formula.<\/p>\n Now the formula for velocity uses displacement or change in location to determine directional advancements.<\/p>\n velocity = \u25b3 distance\u00a0\u00a0\/ time<\/em><\/p>\n Since where you started from is where you've ended, there is no ultimate change in position. This means that the distance covered is 0, and your velocity is also zero.\u00a0If you were traveling in a straight line, the formula for the rate of speed used would equal your velocity.<\/p>\n Most people confuse high speed for the high rate for speed the two are different and distinct. In Physics speed is also referred to as velocity while the rate of speed is known as acceleration. But now that you understand the basics, you won't get them mixed up anymore!<\/p>\nCalculations That Determine the Rate of Speed Formulas<\/h2>\n
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Difference between Average and Instantaneous Speed<\/h3>\n
Calculating Instantaneous Rates of Speed<\/h4>\n
Algebraic Rate of Speed Formulas<\/h3>\n
Rate of Speed Formulas<\/h3>\n
Practice Examples on Rate of Speed Formulas<\/h4>\n
Why You Shouldn\u2019t Confuse Rate of Speed for Velocity<\/h3>\n
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Conclusion<\/h2>\n