ERJ Brainteaser: December

Question 4: Figure out...

Figure out the rubber connection

?, 3, 27, 10

Answer: The key to our final teaser of 2021 - that the numbers are from the periodic table of elements - proved unexpectedly elusive to many readers. So extra well done to the following trio who worked out the starting number as 14 for SI, to give Si Li Co Ne: Frank Bloemendaal, Research & Development, Polycomp, Vorden, The Netherlands: John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK; David Mann, key account manager, SPC Rubber Compounding, UK; and everyone who had a go..

Wishing all our readers - Brainiacs and non-Brainiacs alike - a very Happy Christmas and a great New Year!

Next teaser in early January, when we will also announce the winners in the prestigious Brainiac of the Year Awards!

Question 3: Tennis double

A tennis ball is hit vertically upwards at a speed of 12 m s−1 from a point 60m above the ground. Find: a) the speed of the ball when it strikes the ground; and b) the total time the ball is more than 64m above the ground.

Answer: Quite a few single and double faults for this testing teaser, so extra well done to the trio who aced it (see solutions below): Michele Girardi, quality manager, Scame Mastaf Spa, Suisio, Italy; John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands.

Solutions

Andrew Knox

All you need to know here is that under the acceleration of gravity g, velocity v = g.t, and distance travelled d = 1/2.g.t.t (that is t squared). 

Assuming negligible air resistance effects, the tennis ball decelerates on its way up in the same way it will accelerate on its way down.

a) As velocity v = g.t,  time t on way up to decelerate from speed 12 m/s to 0, or on way down from 0 to 12 m/s, is v/g = 12/9.81 = ca. 1.223 s.

In this time the ball has travelled distance d = 1/2.g.t.t = 9.81/2 x 1.223 x 1.223 = ca. 7.34 m.

So, maximum height reached will be 60 + ca. 7.34 = ca. 67.34 m.

When the ball falls from this height, time t to hitting the ground is therefore given by: t squared = 2 d/g = 2 x 67.34/9.81 = ca. 13.7287, or t = ca. 3.705 s.

Velocity at impact v = g.t = 9.81 x 3.705 = ca. 36.35 m/s.

b) Height travelled by the ball above 64 m is ca.3.34 m. Time spent above this height is therefore twice the time it takes to fall from the maximum height (v=0) to 64 m.

Time to fall 3.34 m is given by t = square root of 2.d/g = root 6.68/9.82 = root 0.68, so t = ca. 0.825 s.

So, total time spent above 64 m = 2 x 0.825 = ca. 1.65 s.

John  Bowen

This one requires use of the Equations of Motion and a bit of algebra.

To calculate the first answer, we need to calculate the ultimate height from which the ball fell - this is where it originally stopped before accelerating earthwards and assumes no effect from air resistance.  

We use vsqd - usqd = 2fs where v = 0 [stopped] u = 12 m/s f = acceleration due to gravity [-9.81m/s/s] and s = distance

so 0 - 144 = 2 x- 9.8 x s,  so s = 7.35m.  As the ball started 60m above the ground, total height it falls through = 67.35m.

Again, using the above formula, from a standstill, vsqd = 2fs, vsqd = 2 x 9,8 x 67.35 = 1320.06, so v = 36.33 m/s

To calculate the time above 64m above the ground, or 4m above its starting point, we need the formula s = ut + 1/2.f.tsqd

so 4 = 12t + 1/2.-9.8.tsqd which rearranges to: 4.9.tsqd - 12t +4 = 0

Solving for t gives values of 0.4 or 2.05 seconds, which means the ball passes the 64 m height upwards at 0.4 seconds, then downwards at 2.05, 

So it is above 64m high for 1.65 seconds.

Michele Girardi

The equation of motion applicable are
Y=Y0+V0*t-1/2.g.t^2          V=V0-g.t

substituting          Y0=60        g=9.8        V0=12

   Y=60+12*t-4.9*t^2

To avoid some boring algebra, we can use an excel file and the solver function to find the solutions for Y equal to 64,64 and zero

point                       t    Y       V
start                       0,0    60,0     12,0
cross 64+              0,4    64,0     8,1
cross 64-             2,1    64,0     -8,1
ground             4,9    0,0     -36,3

delta time     1,7

Question 2: Target number

Using any combination of addition, subtraction, multiplication and/or division, how close to 952 can you get  using the following numbers?

100, 75, 50, 25, 6, 3.

Answer: Expect to see more of this style of question: as well as mathematical wizardry the teaser also gives scope for different solutions and clever approaches (see below). Extra well done to the following readers who were right on or very close to the target figure: John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK; Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Rohit Kalé, distribution strategy manager, AMN/V/B2C/DIS, Michelin North America Inc., USA: Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain; Michele Girardi, quality manager, Scame Mastaf Spa, Suisio, Italy;

Solutions

John Bowen

This needs careful use of brackets:
[100 + 6] x 3 = 318
318 x 75 = 23850
23850 - 50 = 23800
23800 / 25 = 952
Summarised with brackets:  {{ [100 + 6] x 3}  x 75] -50} / 25 = 952 QED as we used to say at good old O-level !

Stephan Paischer

100 x (6+3) + 50 + (75/25) = 
100 x 9 +50 +3 =
900 + 50 +3 = 953

Andrew Knox

100 + 6 = 106
106 x 75 x 3 = 23850
23850 - 50 = 23800
23800 / 25 = 952

Rohit Kalé

((100 + 6) * 3 * 75 - 50) / 25 = 952 

Amparo Botella

100+6=106; 106*3=318; 318*75=23850; 23850-50= 23800; 23800/25=952

Michele Girardi

The answer is 952=1/25*(3*75*(100+6)-50) explanation Let's factor in prime numbers the closest ones
951=3*317
952=2*2*2*7*17
953=prime
954=2*3*3*53
955=5*191
I haven't found combinations for 952
but
952=954-2
=2*3*3*53-2
= 2*3*3*106/2-2
=3*3*106-2
=(3*3*106-2)*25/25
=(3*75*(100+6)-50)/25

Question 1: Latin link

What links the following countries?

Brazil, Peru, Bolivia, Colombia, Ecuador, Venezuela, Guyana.

Bonus question

Using a standard calendar (English-language / Gregorian), multiply the number of months with 28 days by the number of months beginning with a vowel. 

Answers: Not our question-setters’ finest week as French Guiana and Suriname might also have been added to the list of countries linked by the Amazon. Also, the bonus question was a rare exception to our rule of avoiding ‘trick’ questions: all months have 28 days, making the answer 36. To be sporting all round, so, it’s extra well-done this time to everyone who answered one or both questions: Fariha Rashid, marketing analyst, Kraton Polymers LLC, Houston, Texas, USA; Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain; Jose Padron, laboratory analyst, Waterville TG Inc., Waterville, Québec; John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK; Michele Girardi, quality manager, Scame Mastaf Spa, Suisio, Italy; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; David Mann, key account manager, SPC Rubber Compounding, UK; Rohit Kalé, distribution strategy manager, AMN/V/B2C/DIS, Michelin North America Inc. USA: Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria.