1.A force of 40.0 N is needed to compress a spring 0.200 m. A 1.00 x 10-2 kg ball
a) Calculate the work done to compress the spring. (2 marks)
b) What happens to the work done on the spring ? (1 mark)
c) If the spring is released, what happens to the energy of the spring? (1 mark)
d) Calculate the total mechanical energy of the ball at the instant it leaves the spring. (2 marks)
e) What will be the speed of the ball at the instant it leaves the spring? (2 marks)
f) If the ball is fired up into the air by the spring, how much gravitational potential energy will it gain? (1 mark)
g) What will be the maximum height of the ball? (2 marks
5.Exercise 4) A fair coin is tossed. If it lands heads, a fair four-sided die
is thrown (with values 2,3,4,7). If
2,3,4,7). If it lands tails, a fair six-sided die is
thrown (with values 3,4,5,6,7,9). Regardless of which die is used, Alice
eats n grains of rice, where n is the largest prime factor of the die result
(for example, the largest prime factor of 9 is 3).
(a) What is the conditional probability that the coin lands heads, given
that Alice eats three grains of rice?
(b) Suppose that the entire experiment is conducted twice on the following day (starting with a new coin toss on the second run-through).
What is the conditional probability that the coin lands heads on both
run-throughs, given that Alice eats a total of five grains of rice during the two run-throughs?
(Do not count the two grains from part (a) in part (b); we assume
two brand new experiments, each with a new coin toss. Start your
solution by defining a suitable partition of the sample space. Please
use an appropriate notation and/or justification in words, for each
value that you give as part of your solution.)
Exercise 5) Alice and Bob throw an unfair coin repeatedly, with probability 2/5 of landing heads. Alice starts with £2 and Bob starts with £3 .
Each time the unfair coin lands heads, Alice gives Bob £1 . Each time
the unfair coin lands tails, Bob gives Alice £1 . The game ends when one
player has £5 .
(a) Draw a labelled Markov chain describing the problem, and write
down a transition matrix P. Write down the communication classes,
and classify them as either recurrent or transient.
(b) Using the transition matrix, calculate the probability that Alice loses
all of her money in exactly four tosses of the unfair coin.
(c) Calculate the (total) probability that Alice loses all of her money
(before Bob loses all of his).
(d) Calculate the expected (mean) number of tosses of the unfair coin,
for the game to end.
6.We play a variation of Monty Hall: The contestant is presented with 3 curtains, behind each there's a prize with
a prize with probability p independently of the other curtains. The player chooses one curtain
(a) The game host tells the contestant there's a prize hiding behind at least one of the other curtains, and offers the player a chance to change his choice. Should he?
(b) The game host tells the player there's a prize hiding behind exactly one of the other curtains. Should the player change his choice? How does your answer depend on p
(c) How would your answer to section (b) change if there are n curtains in total?
(d) Bonus (4 points): How would your answer to section (a) change if there are 4 curtains in total?
7.Let A be the set that is defined recursively in the following way:
Basis Case: ( 0 , 1 , 2
Recursive Step: if (n,m,r) ∈ A then (n+1,m+2,m+r+2) ∈ A
Prove, using structural induction, that for every .( n , m , r ) ∈ A , 0 ≤ n < m and n + m ≤ r .