A Grasshopper Race Puzzle

This was a Car Talk puzzler from last week. Read the full question on the Car Talk site, but here is the short version.

• Two grasshoppers want to race a distance of 12 feet (there and back).
• Grasshopper A can jump 10 inches at a time.
• Grasshopper B can jump 6 inches at a time.
• At the five foot mark, they are tied. Who wins and why?

I don't want to spoil their puzzler, so I am doing two things. First, I am writing this BEFORE they post the solution. Second, I am not going to post this until AFTER they discuss this on their show. So, the only way it will spoil the puzzle for you is if you are listening to Car Talk via podcasts and you are behind.

So, why am I writing about this puzzle? My first thought was that this involved a little more physics than the usual Car Talk puzzle. Maybe they didn't think this through the whole way (and maybe I didn't think it through the whole way). So, here is my first guess answer (which I will back up with some calculations).

My First Answer: The grasshopper that can jump farther can win (if he changes what he does). Since he can jump 10 inches, he has a greater "launch speed". If he only jumps 6 inches, it will take him less time to travel this distance than it will the 6-inch jumping grasshopper (who will have to jump at a 45 degree angle to get to this distance).

Possible problems with this solution:

• The puzzle states that they are tied after 5 feet. Does this mean that they have different "rejump" times? I suppose that there must be some pause between each sucessive jump.
• I have assumed that the farthest a grasshopper can jump is when jumping at an angle of 45°. Of course, this is only true when air resistance is negligible. I doubt this is a good assumption since the grasshoppers are small (cross sectional area to mass ratio is not constant with size).

Launch Speed vs. Jump Distance

Let me start with the assumption of no (or negligible) air resistance. In this case, the maximum jumping distance happens for a launch angle of 45° (here is a quick derivation of the max range) - oh, this is only true for starting and ending at the same height.

Let me start with a simple diagram.

Yes, I know that trajectory isn't actually a parabola - I was being lazy. The important thing is that I am calling s the range. For this case, the range can be written as:

The cool thing about projectile motion is that the time for the x-motion is the same time for the y-motion. So, here is the y-motion equation for the motion that starts and ends at y = 0 meters.

Using this value for t in the expression for s:

Why? Well, now I know the launch speed of grasshopper A and B. Oh, you like numbers? Ok, if A can jump with an initial speed of 62 inches/s (yes, I don't like those units either but I want to stick with the original puzzle). Grasshopper B has a launch speed of 48 inch/s.

Jumping at Different Angles.

Since the above expression didn't depend on a launch angle of 45°, I can use it to determine the distance and time for a lower angle.

How about this. What is the average velocity for one jump (including rejump time)? I can write that as:

I don't know the time in between jumps (t_r in this case) but I will assume it is constant. Using expressions for s and t from above, I can write this in terms of v₀ and θ.

If the re-jump time is small, then the average velocity just depends on the launch speed and the angle. Of course, there has to be a some re-jump time. Otherwise, the grasshopper could just jump at a 0° angle and basically skip along the ground to win.

Finding the Re-Jump Time

I am going to assume that during the first part of the race both grasshoppers jump at 45° angles (that is sort of implied in the puzzle). The puzzle also says that during this first 5 feet, grasshopper B jumps 10 times (they call him Rocky) and A has only jumped 6 times. Since they have the same average velocity, I can write this as:

I know I shouldn't have done this, but I switched labels. I am calling v_A the jump speed of grasshopper A - is that ok? Anyway, I know the launch speeds and I know g = 386 inch/s². So, I can write this as:

Yes, I skipped some steps in the algebra - sorry. But what does this say? It says that in order for the two grasshoppers to tie at 5 feet, the re-jump time for grasshopper B would have to be less than half the re-jump time compared to grasshopper A.

A Model

Let me jump to a model. First, I will pick a re-jump time for grasshopper A with a value of 0.2 seconds (randomly determined). Here is a plot of the position vs. time for both of these jumping grasshoppers if they are both jumping at 45° launch angle.

There are two things to notice in this plot. First, grasshopper A (the one that can jump 10 inches and the blue line) has a faster horizontal speed during the jump as well as fewer jumps. Second, the only way for grasshopper B to be even with A, it has to take much shorter pauses in between jumps.

Ok, now let me let grasshopper B jump at a 30° angle. Here is a plot of their positions for a short time for the first 12 feet.

Here, grasshopper B can win. How is this possible? Well, since B has a such a short re-jump time, he (I am assuming a male grasshopper from the Car Talk problem) can take more and shorter jumps. For the shorter jumps, his average velocity is higher.

What happens if A jumps at at 30° angle and B at a 45° angle? Here is that plot.

Grasshopper A doesn't really gain an advantage. Why? Because even though his average speed during the jump is greater, he has more jumps. For whatever reason, his re-jump time is too high for this to be an advantage.

But what angle is the best launch angle? Here is my last plot (really). This is the average speed over one jump (with waiting time) for both grasshoppers as a function of jump angle.

There is a slight problem. These two curves should have the same average speed at 45°. I will blame it on a rounding error (but I am not absolutely sure). However, this does show the point I am trying to make. For the grasshopper B (green curve), he can really increase his average speed by taking shorter jumps. His optimal angle seems to be around 30° (I just guess at this before). But for grasshopper A, he really won't get too much benefit from shorter jumps since his in-between jump time is too large.

Conclusion

I think my initial answer was correct except that I had the wrong grasshopper. The shorter jumping grasshopper could win if he jumped at an angle lower than 45°. The other important conclusion is that I am pretty sure this puzzle was WAY more complicated than Tom and Ray (from Car Talk) had intended. Or maybe there is a simpler solution and I am just over thinking things.

Either way, I had fun with this problem. Also, I should get some type of bonus award for "taking it too far" or something.

AH. Air resistance. I forgot to consider air resistance. Well, maybe I can save that for another post.

Car Talk Solution

I just looked at the solution on Car Talk. Essentially, their answer is legit. They say the short jumping grasshopper wins since his jumps fit an integer number of times in 12 feet (the distance to the turn around point). The other grasshopper will jump past the turn around point and take longer to finish. Ok, this is a fine solution IF you assume the grasshoppers can't change how far they jump. I know several grasshoppers personally. They can all change the distance of their jump. So there.