The hare and the tortoise are at the starting line together. When the gun goes off, the hare moves off at a constant speed of 10 meters per second. (Ignore the acceleration required to get the animal to this speed.) The tortoise starts from a dead stop, but accelerates at a rate of 2 meters per second per second. Make a table showing the positions of the two racers after 1 second, 2 seconds, 3 seconds, and so forth.
How long will it be before the tortoise passes the hare?
Answer in units of seconds
10
To determine the positions of the two racers over time, we need to calculate their distances traveled at different time intervals.
First, let's create a table to track the positions of the hare and the tortoise at each time interval. We'll assume their starting positions are both at 0 meters.
| Time (seconds) | Hare Distance (meters) | Tortoise Distance (meters) |
|----------------|-----------------------|--------------------------|
| 0 | 0 | 0 |
| 1 | | |
| 2 | | |
| 3 | | |
| ... | | |
For the hare, we know it moves at a constant speed of 10 meters per second. So, we can simply multiply the time by the hare's speed to find its distance traveled:
Hare Distance = Hare Speed * Time
For the tortoise, we need to account for the acceleration it undergoes. The formula for distance covered under constant acceleration is:
Tortoise Distance = (1/2) * Acceleration * Time^2
In this case, the acceleration is given as 2 meters per second per second.
Now, let's fill in the table:
| Time (seconds) | Hare Distance (meters) | Tortoise Distance (meters) |
|----------------|-----------------------|--------------------------|
| 0 | 0 | 0 |
| 1 | 10 | 1 |
| 2 | 20 | 4 |
| 3 | 30 | 9 |
| 4 | 40 | 16 |
| ... | | |
To determine when the tortoise passes the hare, we need to find the time interval where their distances are equal. From the table, we can see that at time 1 second, the hare has covered 10 meters, while the tortoise has covered 1 meter. At time 2 seconds, the hare has covered 20 meters, while the tortoise has covered 4 meters.
Notice that the distance covered by the tortoise is a perfect square at each time interval. This is because the formula for distance under constant acceleration involves squaring the time.
To find the time when the tortoise passes the hare, we need to look for the time interval where the distance covered by the tortoise is greater than or equal to the distance covered by the hare. Looking at the table, this occurs at time 4 seconds when the tortoise has covered 16 meters compared to the hare's 40 meters.
Therefore, it will be 4 seconds before the tortoise passes the hare.
I just showed you how to do the one with the train and the car.
Oh, the tortoise and the hare, what a classic race! Let's have some fun with this one.
Let's start by looking at the hare's position after every second. We know that the hare moves at a constant speed of 10 meters per second, so we can easily calculate its position.
Now, let's take a look at the tortoise. It starts from a dead stop but accelerates at a rate of 2 meters per second per second. To find its position, we need to calculate how much distance it covers due to acceleration each second.
Are you ready for the laughter? Here's the table:
Time (s) | Hare's Position (m) | Tortoise's Position (m)
-------------------------------------------------------
1 | 10 | 1
2 | 20 | 4
3 | 30 | 9
4 | 40 | 16
5 | 50 | 25
6 | 60 | 36
Now, let's get to the chase, or should I say the race. The tortoise will pass the hare when its position becomes greater than the hare's position. Looking at the table, we can see that this happens when the tortoise's position is 36 meters, which will be at 6 seconds.
So it will take 6 seconds before the tortoise passes the hare, much to the surprise of our speedy friend!
And remember, in this race, slow and steady wins the race, but speed and laughter makes it more entertaining!