Motion (student)

WORKSHEET - Mathematics (Advanced) 14. Applications of Calculus to the Physical World ' ] Motion ' Teacher: James Butterworth Exam Equivalent Time: 159 minutes (based on HSC allocation of 1.5 minutes approx. per mark) The Big Picture: Calculus in the Physical World 16.0% 14.0% 12.0% i 10.0% | - 8.0% | " F J N H . 4.0% e i | : 1 B B 2.0% 0.0% % HSC Exam ™ T2 T3 T4 T5 T6 T7 T T10 ™ T12 T13 T14 *SmarterMaths analytics based on the last 10 years' HSC exams. M T14 - Growth and Decay M T14 - Motion i T14 - Rates of Change = ® Other Topics HISTORICAL CONTRIBUTION o T14 Calculus in the Physical World has contributed a substantial 9.8% to the HSC exam, on average, over the past 10 years. ¢ This topic has been split into three sub-categories for this analysis which are: 7-Motion (4.4%), 2-Exponential Growth and Decay (4.1%), and 3-Rates of Change (1.3%). 2017 HSC ANALYSIS - What to expect and common pitfalls o Motion (4.4%)is examined every year. 2016 saw a significant allocation of 8 marks, while 2013-2015 tested the area twice each year through a multiple choice and longer answer questions. e Longer answer Motion questions have used equations that are a simple function of ¢ for the last 4 consecutive years. The 4 years prior to that, the equations of motion were trig ore based functions that therefore warrant attention. ¢ Although Motion questions have proven challenging for a majority of students, they consistently offer up much easier marks in earlier parts. ¢ ltis worth taking particular note of the poorly answered multiple choice question in 2013 (Q10). Questions 1. Calculus in the Physical World, 2UA 2014 HSC 9 MC The graph shows the displacement & of a particle moving along a straight line as a function of time . XA ~Y O Which statement describes the motion of the particle at the point P? (A) The velocity is negative and the acceleration is positive. (B) The velocity is negative and the acceleration is negative. (C) The velocity is positive and the acceleration is positive. (D) The velocity is positive and the acceleration is negative.

2. Calculus in the Physical World, 2UA 2017 HSC 10 MC 4. Calculus in the Physical World, 2UA 2015 HSC 9 MC A particle is moving along a straight line. A particle is moving along the &-axis. The graph shows its velocity ¥ metres per second at The graph shows the velocity, v, of the particle fortime £ > 0. time ¢ seconds. v vi A 8 ~Y < How many times does the particle change direction? 0 4\ T (A) 1 (B) 2 €3 When t = 0 the displacement z is equal o 2 metres. (D) 4 What is the maximum value of the displacement 2? (A) 8 m 3. Calculus in the Physical World, 2UA 2013 HSC 10 MC (B) 14m A particle is moving along the z-axis. The displacement of the particle attime ¢ seconds is (C) 16 m x metres. D) 18m Atacertaintime, £ = — 3ms' and & = 2ms™. Which statement describes the motion of the particle at that time? (A) The particle is moving to the right with increasing speed. (B) The particle is moving to the left with increasing speed. (C) The particle is moving to the right with decreasing speed. (D) The particle is moving to the left with decreasing speed.

5. Calculus in the Physical World, 2UA 2008 HSC 6b 6. Calculus in the Physical World, 2UA 2015 HSC 14a The graph shows the velocity of a particle, ¥ metres per second, as a function of time, ¢ In a theme park ride, a chair is released from a height of 110 metres and falls vertically. seconds. Magnetic brakes are applied when the velocity of the chair reaches —37 metres per second. 20 Lt 110m ) T" l , I E : ? ———— Magnetic brakes NOT TO o 2 4 6 b 12 4 SCALE (i) Whatis the initial velocity of the particle? (1 mark) (i) When is the velocity of the particle equal to zero? (1 mark) The height of the chair at time ¢ seconds is £ metres. The acceleration of the chair is given _ _ _ by & = — 10.Atthe release point, t = 0,2 = 110 and & = 0. (iif) When is the acceleration of the particle equal to zero? (1 mark) i i = — (iv) By using Simpson's Rule with five function values, estimate the distance travelled by () Using calculus, show that 2 5% + 110. (2 marks) the particle between ¢ = 0 and ¢t = 8. (3marks) (ii) How far has the chair fallen when the magnetic brakes are applied? (2 marks) 7. Calculus in the Physical World, 2UA 2014 HSC 13c The displacement of a particle moving along the z-axis is given by A s where 2 is the displacement from the origin in metres, ¢ is the time in seconds, and ¢ > 0. (i) Show that the acceleration of the particle is always negative. (2 marks) (ii) What value does the velocity approach as ¢ increases indefinitely? (1 mark)