2020HSCShort answer

Q12 (b) (iii)

2 marks

Question

Given $u^2>gR$ , prove that there are two distinct values of $\theta$ for which the particle will land at $x=R$ .

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	2
Attempts to use the discriminant $\Delta=\frac{4u^4}{g^2x^2}-4$ , or equivalent merit	1

Sample answer:

From part (ii), we require the quadratic to have two distinct real roots, when $x=R$ .

\Delta=\frac{4u^4}{g^2R^2}-4=\frac{4(u^4-g^2R^2)}{g^2R^2}

Given $u^2>gR\Rightarrow u^4>g^2R^2$ .

\therefore\Delta>0

Hence there are two distinct values for $\tan\theta$ , hence two distinct values for $\theta$ , as $\tan\theta$ is increasing.

Supplied marking guidance · human reviewed

Compare this with your working, not only your final sentence.

Syllabus alignment

Working MathematicallyMechanicsProof

The Nature of ProofUse formal proof language, contradiction, counter-examples and rigorous arguments involving numbers and inequalities.
Projectiles and resisted motionUse parametric and Cartesian equations to solve projectile problems with and without resistance.

Spot a problem? Send feedback