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In physics speed, distance and time are three basic parameters that can be used to solve many problems, if we know how to relate them. Distance is the space covered by a moving object or the length between two points. The letter d is generally used in formulas and equations to identify the distance. Speed is the distance that an object or a person travels in a given period of time. Usually the letter v is used to identify speed. Time is the measured or measurable period during which an action or process develops, and it is identified with the letter tin formulas and equations. In the problems that relate to distance, speed and time, time is considered as the specific period in which a certain distance is covered.
How to write problems involving speed, distance and time
When posing a problem involving speed, distance, and time, it will be helpful to organize the information into diagrams or graphs. The formula that relates these three parameters is the following: distance = speed x time . And it is expressed using the symbols of each parameter:
d=vt
There are many simple real life examples where this formula can be applied. For example, in the case of a person traveling on a train, if you know the time the person traveled and the average speed of the train, you can easily calculate the distance the person traveled. And if you know the time and distance an airplane passenger traveled, the average speed of the plane can be calculated by reconfiguring the above formula.
Examples of problems involving speed, distance, and time
In general, a problem of this type asks a question about one of the three parameters, knowing the remaining two, and it is solved with a simple arithmetic calculation substituting the values in the formula.
For example, suppose a train leaves a certain location and travels at 50 kilometers per hour (km/h) (train 1). Two hours later, another train leaves the same place (train 2) traveling on a track adjacent or parallel to the first train but traveling at 100 km/h. How far from the starting point will the faster train catch up with the slower train?
To solve the problem, we define d as the distance in kilometers that each train travels from the starting point until it meets, and as t the time it takes the slowest train to travel that distance. It may be helpful to diagram the problem to better visualize it. The formula we will use is:
distance = speed x time
When posing a problem, the units of the parameters available to solve it must be clearly indicated. Distance can be expressed in meters or kilometers, and time in seconds, minutes, or hours. The units of the speed will be the combination of the units of distance and time, since it is defined as the distance traveled over a certain time; They can be meters per second (m/s), kilometers per hour (km/h), or any other combination.
Let’s see how to solve the problem with the equation that relates speed, distance and time. The condition that arises is that the two trains have traveled the same distance. The distance traveled by each train has the following expression:
train 1 d=50.t
train 2 d=100.(t – 2 )
Keep in mind that train 2 leaves 2 hours later than train 1; therefore, the time it travels is that of train 1, which we define as t , minus 2 hours.
According to the stated condition that they travel the same distance, we can equate both expressions
50.t=100.(t – 2 )
and from this equation clear the value of t . To do this, we divide both terms of the equality by 50 and develop the factor in parentheses, and we obtain:
t=2t – 4
Solving for the value of t, it is obtained that the time required for train 2 to catch up with train 1 is 4 hours. If this value of time is substituted in the distance expression for train 1, it is obtained that both trains meet after traveling 200 km.
Let’s look at another example. A train left Lima for Huancayo. Five hours later, another train also left for Huancayo, traveling at 40 km/h with the aim of catching up with the first train. The second train finally caught up with the first after traveling for three hours. What is the speed of the train that left first? This problem is similar to the first, but both the information available and what you want to find out is different. Let’s set up the equations corresponding to both trains, but now we want to find out the speed v of train 1, and we consider that the time t is the time that train 2 travels, since that is one of the data.
train 1 d=v.(3+5)
train 2 d=40.(3 )
By equating both expressions since both trains travel the same distance, it is obtained that
8 . v=120
with which we obtain, by dividing both terms of the equality by 8, that the speed v of the first train was 15 km/h.
Let’s see a third example, also with trains. A train (train 1) left the station and traveled to its destination at 65 km/h. Later another train (train 2) left the station traveling in the opposite direction of the first train at 75 km/h. After having traveled for 14 hours, the first train was at a distance of 1,960 km from the second train. How long did the second train travel? As in the previous cases, let’s formulate the equations corresponding to both trains, but now our unknown is the time t that train 2 traveled.
train 1 d=65.(14)
train 2 d=75.t
In this case, the relationship between both equations is that the sum of the distances traveled by each train is 1960 km, since they depart in opposite directions. This relationship is expressed in the following equation:
65.(14) + 75.t = 1960
910 + 75.t = 1960
Subtracting 910 from each equality term
75.t = 1050
And dividing both terms by 75 we have that the time that the second train travels is 14 hours, just like the first train.
