Equations Representing the Relationship Between Distance, Time and Velocity

The Idea

One car is going faster than the other, but the slower car has a head start. We can predict where and when the faster car will overtake the slower car. All we have to do is graph the movement of each car and see where the lines cross. This experiment gives you a method to make that prediction.

What You Need

• 2 toy cars with adjustable speeds
• stopwatch
• tape measure

Method

1. Set the speed of each of the two cars, so one is faster than the other. (If you don't know the speeds before starting, you can measure them in the following steps.)
2. Determine the average velocity of each of the cars by measuring the distance it goes in a given amount of time. The equation is average velocity = (distance traveled) divided by (time to get there). In physics, meters are typically used to measure distance (to be consistent with the SI or System International unit system). This will result in velocity measured in meters per second (m/s). However, you can work with other units for distance (such as feet per second) as long as you are consistent.
3. Line up the two cars in the same direction on a level floor heading in the same direction, as shown in Figure 3-1.
4. We are going to give the slower car a head start of a few seconds and try to predict where the faster car will overtake the slower car.
5. To do this:
   • Plot the speed of the faster car on a graph of distance versus time with the line starting at the origin and having a slope equal to the speed of the faster car.
   • Plot the speed of the slower car on the same graph, but starting at a point where the distance is zero and the time is equal to the chosen time delay.
   • See Figure 3-2, which shows a slower car going at 0.25 meter per second car given a 0.25 meter head start in front of a faster car going 0.4 meter per second. (Notice the slower car is predicted to overtake the faster car at a point that is 0.68 meters from the starting point and 1.8 seconds after the race starts.)





6. Predict where the faster car you are working with will overtake the slower car.
7. Start the slower car and give it a head start.
8. Compare where and when the faster car will overtake the slower car with your predictions.

Expected Results

The faster car will overtake the slower car when the two lines in the graph cross. The distance the lines cross at is how far from the starting line the faster car catches the slower car.

The time where the lines cross is how many seconds from the start of the race when the slower car catches the faster car.

Why It Works

The distance that a object goes is given by the equation:

where d_° is the initial distance between where the object starts and the starting line. (d_° can be understood as the head start in distance) v is the velocity of the car

t is the time it has been going from the start of the race, and t_° is the delay or the head start in seconds given to the other car.

Other Things to Try

Here are some alternative ways of doing this:

1. If you have two motion sensors, focus one on the faster car and the other on the slower car. This generates a similar curve as shown in Figure 3-2. If the cars are moving away from you a similar curve will be produced, except the slope will be positive.
2. Another way to establish two different velocities is to use objects rolling off two different slopes starting from different heights. The object starting from the higher starting point will be rolling on the table or floor with a higher velocity, with the velocity proportionate to the height difference. If the bottoms of each of the ramps are the same distance from the starting line, the slower rolling object can be given a few seconds head start. A similar prediction and comparison of results can be made as in the previous section.
3. If you happen to be associated with a FIRST robotics team, you may want to consider using last year's robot(s) for this experiment.
4. Another variation is to predict where and when two cars moving toward each other will meet.

The Point

Two objects that move independently can be represented by separate equations that represent the relationship between distance and time. These are two simultaneous equations, which can be solved graphically to find the time and distance that the faster object overtakes the slower object.

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