Distance, Velocity and Time: Equations and Relationship
Do you remember the story of the tortoise and the hare? The tortoise and the hare decided to have a race. Everybody, most of all the hare, thought the hare would win the race because he hopped much faster than the tortoise plodded. We could say that the hare usually travelled at a much higher velocity than the tortoise. Velocity is the measure of the amount of distance an object covers in a given amount of time. Here's a word equation that expresses the relationship between distance, velocity and time: Velocity equals distance travelled divided by the time it takes to get there.
Confident that his high typical velocity would allow him to cover the race distance in a short amount of time, the hare took a lot of breaks. Looking at our equation, we can see that when time is increased—for any reason—overall velocity goes down. The tortoise never took any breaks: he just kept plodding along. By the time the hare realized that he was way behind the tortoise, it was too late. The tortoise won the race, and we all got the moral “Slow and Steady Wins the Race.” That’s a good lesson but the hare could have won the race (and still taken a few breaks) if he had only completed the experiment you are about to do.
Problem: Predict when a fast toy car can pass a slower toy car.
- A smooth, clear, level stretch of floor (two 2-meter stretches of flat Hot Wheels track is even better)
- 2 toy cars with adjustable speeds
- Tape measure
- Stopwatch or timer
- Graph paper
- Use your tape measure to mark off a distance of 1.5 meters from the starting line.
- Set one of your cars to a “slow” speed. Have your assistant start the timer when set the car down at the starting line.
- Watch the car carefully. It should travel in a straight line and at a constant velocity.
- When the slower car passes the 1.5 meter mark, have your assistant stop and record the time.
- Repeat this a second or third time to make sure you obtain a consistent velocity across all trials.
- Divide distance by time to determine the slower car’s velocity in meters per second.
- Set your second car to a faster speed than your first.
- Using your calculated velocities, you will now are going to determine how long your faster car could delay and still overtake the slower car.
- On the X axis of your graph paper, mark the time in ½ second increments from zero to two seconds beyond the time it took your slower car to complete the course. Make sure to make your graph large so you have room to expand it if necessary!
- On the Y axis, mark the distance in centimeters from zero to 150 centimeters. (Remember—150 centimeters is the same as 1.5 meters.)
- Plot the velocity of the slower car. Your first point should be at (0,20 cm) because you are going to give it a 20-cm head start.
- Your second point for the slow car is the velocity you measured. The X value should be whatever time it took for the car to reach the end of your test course, your Y value is the distance you had the car travel (1.5 meters).
- Using your ruler and pencil, connect the two points to make a line.
- Now, plot the velocity of the faster car. Your first point should be at (0,0 cm) because this car will not get a head start.
- Your second point for the fast car is the velocity you measured. The X value should be whatever time it took for the car to reach the end of your test course, your Y value is the distance you had the car travel (1.5 meters).
- Using your ruler and pencil connect the two points to make another line. Make this line look different than the first, either by making dashes or making it darker.
- Label the lines fast car and slow car.
- Find where the two lines cross. At this intersection point, trace one line to X axis, and another to the Y axis. These are the lines with arrows on diagram 1. The two values you see are the time and distance where the fast car should overtake the slower car. If your lines don’t cross in your graph, you need to shorten the slow car’s head to start to 10 or even 5 centimeters.
- Mark the predicting passing point on your course.
- Now, it’s time to test your prediction!
- Mark off the slow car’s head start point on your course.
- Mark off the calculated point where the faster car should overtake the slower car.
- Have your assistant release the slower car at the head start mark while you simultaneously release your faster car at the starting line. Start the timer (a third person might be nice for this).
- Watch carefully to see where the fast car overtakes the slow car.
- Compare your predicted time and distance that the fast car overtook the slower car with the actual values.