SSCE HSC Mathematics Extension 1 Bernoulli trials: A projectile is fired from the o

Question

A projectile is fired from the origin O with initial velocity V ms⁻¹ at an angle θ to the horizontal. The equations of motion are given by

$x = V 	ext{cos} θ, \quad y = V \text{sin} θ - \frac{1}{2} g t^2.$  
(Do NOT prove this.)

(i) Show that the horizontal range of the projectile is $ \frac{V^2 \sin 2θ}{g} $.

A particular projectile is fired so that θ = $ \frac{π}{3} $.

(ii) Find the angle that this projectile makes with the horizontal when  $ t = \frac{2V}{\sqrt{3g}} $.

(iii) State whether this projectile is travelling upwards or downwards when  $ t = \frac{2V}{\sqrt{3g}} $. Justify your answer.

(b) A particle is moving horizontally. Initially the particle is at the origin O moving with velocity 1 ms⁻¹.  
The acceleration of the particle is given by $ \ddot{x} = x - 1, $ where x is its displacement at time t.

(i) Show that the velocity of the particle is given by $ \dot{x} = e^{t - x}. $

(ii) Find an expression for x as a function of t.

(iii) Find the limiting position of the particle.

(c) Two players A and B play a series of games against each other to get a prize. In any game, either of the players is equally likely to win.  
To begin with, the first player who wins a total of 5 games gets the prize.

(i) Explain why the probability of player A getting the prize in exactly 7 games is $ \binom{6}{1} \left( \frac{1}{2} \right)^7 $.

(ii) Write an expression for the probability of player A getting the prize in at most 7 games.

(iii) Suppose now that the prize is given to the first player to win a total of (n + 1) games, where n is a positive integer.

By considering the probability that A gets the prize, write a formula for $ \binom{2n}{n} $.

A projectile is fired from the origin O with initial velocity V ms⁻¹ at an angle θ to the horizontal - HSC - SSCE Mathematics Extension 1 - Question 14 - 2015 - Paper 1

Worked Solution & Example Answer:A projectile is fired from the origin O with initial velocity V ms⁻¹ at an angle θ to the horizontal - HSC - SSCE Mathematics Extension 1 - Question 14 - 2015 - Paper 1

Show that the horizontal range of the projectile is \( \frac{V^2 \sin 2θ}{g} \)

Find the angle that this projectile makes with the horizontal when \( t = \frac{2V}{\sqrt{3g}} \)

State whether this projectile is travelling upwards or downwards when \( t = \frac{2V}{\sqrt{3g}} \). Justify your answer.

Show that the velocity of the particle is given by \( \dot{x} = e^{t - x}. \)

Find an expression for x as a function of t.

Find the limiting position of the particle.

Explain why the probability of player A getting the prize in exactly 7 games is \( \binom{6}{1} \left( \frac{1}{2} \right)^7 \).

Write an expression for the probability of player A getting the prize in at most 7 games.

By considering the probability that A gets the prize, write a formula for \( \binom{2n}{n} \).

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