# Determining Rate of Speed Formulas

By Kathleen Cantor, 07 Oct 2020

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! We'll walk you through determining the rate of speed formulas in algebraic calculus.

## What Is Rate of Speed?

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 longer distances than the one moving slowly, if the time they're given is the same.

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 involve the distance traveled and time taken.

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.

## Calculations That Determine the Rate of Speed Formulas

Speed is related to velocity, and that’s 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;

• When time (t) is constant, speed is directly proportional to distance. v ∝ s (t)
• When distance (s) is constant, speed is inversely proportional to time. v ∝ 1/t (s)

To give a symbolic form definition of speed, combine these two rules to derive; v = s/t

Though this is not the final definition, you can also define speed as the rate of change in distance within a specified time.

### Difference between Average and Instantaneous Speed

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.

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.

We'll use this formula:  v = s/t

≈ 200 m/4 hour = 50 m/h

That's the equation's product, but is the speed of 50 miles per hour accurate for the entire 4-hour journey?

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 average rate of speed for the trip.

You can modify the speed rate equation to further emphasize this:

¯v =∆s/ ∆t

Look at the bar above the speed symbol (¯v), which indicates that you are calculating mean or average. The delta (∆) symbol represents the change made or distance covered.

#### Calculating Instantaneous Rates of Speed

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.

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 effectively calculate instantaneous speed by writing it symbolically:

v = lim     ∆s/∆t  = ds/dt

∆t→0

In calculus, you can read this formula as speed being the first derivative of distance traveled with respect to the time taken.

### Algebraic Rate of Speed Formulas

When determining the rate of speed formulas in algebra, the most commonly used expression is as follows:

distance = rate x time.

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.

Rate = distance  / time

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.

The speed rate can be written as a fraction with a denominator of time and a numerator of distance units. An 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.

Expressing the units must remain true when using the formula rate of speed x time = distance. If 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.

30  miles/hour   x 2 hours = 60 miles

The hours will cancel out and leave only miles.

### Rate of Speed Formulas

Rate is a type of ratio, and as such, is used in many everyday scenarios,  including grocery shopping, medicine, and traveling. Many activities will involve some form of rate. The heart rate is the number of heartbeats per second, while a babysitter will charge a specific rate which is a dollar amount per hour.

To determine the speed formulas rate, you'll use rates such as feet per second or miles per hour.  You can find the clue that you're dealing with a rate when you come across the word 'per.'

In mathematical problems, the word per is replaced by a forward dash (/). When determining the rate of speed formulas, the rate is equal to the distance over time. The formula can be written like this:

r (rate of speed) = d (distance) / t (time)

Or you can use (d = rt) for solving distance and (t=d/r) for time. This is a formula derived from cross products and proportions, and one step of the multiplication has been handled for you.

#### Practice Examples on Rate of Speed Formulas

If you rode a bike for 24 miles and spent 2 hours on the road, calculate your speed rate?

Using the r=d/t formula, your speed rate becomes 24miles divided by the 2 hours, which equals 12 miles per hour.

r = 24 miles / 2 hours = 12 mph

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:

d = 10 mph x 4 hours = 40 miles

If you were to travel at a speed of 12 miles per hour for 18 miles, how much time did you take? Use the t=d/r formula:

t = 18 miles / 12 mph = 1 ½ hours

### Why You Shouldn’t Confuse Rate of Speed for Velocity

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 change of an object's position over time.

Let's say you drive from your home to your workplace and back again within an hour for the 20 miles distance. Calculate your average speed.

Since you’ve 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.

• Speed time/distance
• Speed = 40/1= 40mph

Now the formula for velocity uses displacement or change in location to determine directional advancements.

velocity = △ distance  / time

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. If you were traveling in a straight line, the formula for the rate of speed used would equal your velocity.

## Conclusion

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!

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