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Mechanics

Apply calculus to motion problems: use all forms of acceleration (

\frac{dv}{dt}

,

v\frac{dv}{dx}

,

\frac{d}{dx}(\frac{1}{2}v^2)

); analyse simple harmonic motion including derivation of displacement and velocity; solve motion problems with resistance proportional to velocity or

v^2

, including terminal velocity and projectile motion with air resistance.

Mechanics · MEX-12-04

Sub-topics

Velocity, acceleration and calculus

Express acceleration in all three equivalent forms:

\frac{dv}{dt}

v\frac{dv}{dx}

\frac{d}{dx}\!\left(\frac{1}{2}v^2\right)

; solve motion problems by integrating equations of motion with appropriate initial conditions; choose the most efficient form for a given problem.

Simple harmonic motion

Recognise and solve the SHM equation

\ddot{x} = -n^2 x

; derive the velocity formula

v^2 = n^2(A^2 - x^2)

a = v\frac{dv}{dx}

; write displacement in the form

x = A\sin(nt + \alpha)

x = A\cos(nt + \alpha)

; find amplitude, period, maximum speed and maximum acceleration; solve SHM problems in context.

Resisted motion and projectiles

Model motion with resistance proportional to velocity (

R = mkv

v^2

R = mkv^2

); solve the resulting separable differential equations; find and interpret terminal velocity; model projectile motion incorporating air resistance.

Open Math

Mechanics

Mechanics · MEX-12-04

Name:

Date:

Velocity, acceleration and calculus

Q1Straightforward

A particle moves along a straight line so that its velocity at time

t

seconds is

v = 3t^2 - 2t

m/s. Find the acceleration of the particle at

t = 3

s.

Q2Straightforward

A particle moves with acceleration

a = 6t + 4

m/s². Given that

v = 2

m/s when

t = 0

, find the velocity when

t = 3

s.

Q3Straightforward

A particle starts from rest at the origin with acceleration

a = 2t

m/s². Find its displacement (in metres) from the origin when

t = 4

s. Give your answer to two decimal places.

Q4Straightforward

A particle moves so that

v = 3x^2

m/s, where

x

is displacement in metres. Find the acceleration when

x = 2

m, using

a = v\dfrac{dv}{dx}

.

Q5Moderate

A particle moves with acceleration

a = 2x

m/s² (where

x

is displacement). Given that

v = 0

when

x = 0

, find the speed when

x = 3

m. Give your answer to two decimal places.

Q6Moderate

A particle has acceleration

a = -4x

m/s² and starts with

v = 6

m/s at

x = 0

. Find its speed when

x = 1

m. Give your answer to two decimal places.

Q7Moderate

A particle starts from rest at

x = 1

with acceleration

a = \dfrac{2}{x^2}

m/s². Find its speed when

x = 2

m. Give your answer to two decimal places.

Q8Moderate

A particle has displacement

x = t^3 - 6t^2 + 9t

m for

t \geq 0

. Find the total distance travelled from

t = 0

to

t = 3

s.

Q9Moderate

A particle has acceleration

a(v) = 6 - 2v

. Find the terminal velocity (the limiting velocity as

t \to \infty

).

Q10Challenging

A particle moves from rest with acceleration

a = 4 - 2v

m/s² (where

a > 0

initially). Find the time (in seconds) taken to reach

v = 1

m/s. Give your answer to two decimal places.

Q11Challenging

A particle has acceleration

a = v^2

m/s² and initial velocity

v = 1

m/s at

t = 0

. Find the velocity when

t = 0.5

s.

Q12Straightforward

A particle has acceleration

a = e^{-t}

m/s² and starts from rest at

t = 0

. Find the velocity (in m/s) at

t = 2

s. Give your answer to two decimal places.

Q13Challenging

Prove that the acceleration of a particle moving in a straight line can be written as

a = v\dfrac{dv}{dx} = \dfrac{d}{dx}\!\left(\dfrac{1}{2}v^2\right)

, where

v

is velocity and

x

is displacement.

Simple harmonic motion

Q14Straightforward

A particle undergoes simple harmonic motion satisfying

\ddot{x} = -9x

. State the value of

n

, where

\ddot{x} = -n^2 x

.

Q15Straightforward

A particle in SHM has displacement

x = 4\sin(3t)

m. State the amplitude (in metres).

Q16Straightforward

A particle in SHM has displacement

x = 4\sin(3t)

m. Find the period (in seconds). Give your answer to two decimal places.

Q17Straightforward

A particle moves in SHM with

x = 3\cos(2t)

m. Find its maximum speed (in m/s).

Q18Moderate

A particle moves in SHM with amplitude

5

m and period

4

s. Find the speed (in m/s) when

x = 3

m. Give your answer to two decimal places.

Q19Moderate

A particle satisfies

\ddot{x} = -16x

and starts from rest at

x = 3

m. Find its speed (in m/s) when

x = 1

m. Give your answer to two decimal places.

Q20Moderate

A particle has displacement

x = 5\sin\!\left(2t + \dfrac{\pi}{6}\right)

m. Find the velocity (in m/s) at

t = 0

. Give your answer to two decimal places.

Q21Straightforward

A particle satisfies

\ddot{x} + 25x = 0

. Find the period of its motion (in seconds). Give your answer to two decimal places.

Q22Moderate

A particle in SHM has period

\dfrac{\pi}{2}

s and amplitude

3

m. Find the maximum acceleration (in m/s²).

Q23Challenging

A particle starts at

x = 0

moving in the positive direction with SHM:

n = 2

, amplitude

A = 4

m. Find the first positive time

t

(in seconds) at which

x = 2

m. Give your answer to two decimal places.

Q24Challenging

A particle in SHM has amplitude

10

m and maximum speed

5

m/s. Find the period (in seconds). Give your answer to two decimal places.

Q25Straightforward

A particle in SHM has

x = 3\cos(4t)

m. Find the acceleration when

t = \dfrac{\pi}{8}

s.

Q26Challenging

A particle moves in SHM satisfying

\ddot{x} = -n^2 x

. Prove that

v^2 = n^2(A^2 - x^2)

, where

A

is the amplitude (the maximum displacement from the centre).

Resisted motion and projectiles

Q27Straightforward

A particle of mass

2

kg moves horizontally with a resistance force

R = 4v

N (opposing motion). Given initial velocity

v_0 = 10

m/s, find the velocity (in m/s) at

t = 1

s. Give your answer to two decimal places.

Q28Straightforward

A particle satisfies

\dot{v} = -3v

with

v(0) = 12

m/s. Find the velocity (in m/s) when

t = 0.5

s. Give your answer to two decimal places.

Q29Straightforward

A particle of mass

2

kg falls from rest under gravity with air resistance

R = 2v

N. Using

g = 10

m/s², find the terminal velocity (in m/s).

Q30Straightforward

A particle of mass

1

kg falls with air resistance

R = 0.1v^2

N (taking downward as positive,

g = 10

m/s²). Find the terminal velocity (in m/s).

Q31Moderate

A particle of mass

1

kg falls from rest with equation of motion

\dot{v} = 10 - 2v

(taking downward as positive,

g = 10

m/s²). Find the velocity (in m/s) at

t = 1

s. Give your answer to two decimal places.

Q32Moderate

A

5

kg particle moves horizontally with resistance

R = 5v^2

N and no driving force. At

t = 0

,

v = 20

m/s. Find the time (in seconds) when

v = 10

m/s. Give your answer to three decimal places.

Q33Moderate

A particle of mass

1

kg falls from rest with equation of motion

\dot{v} = 10 - 0.5v

(downward positive). Find the distance fallen (in metres) in the first

2

s. Give your answer to two decimal places.

Q34Straightforward

A falling particle satisfies

\dot{v} = 10 - 0.2v

(with

g = 10

m/s²). Find the terminal velocity (in m/s).

Q35Moderate

A particle of mass

1

kg moves horizontally with a resistance force

R = 0.1v^2

N. Its initial velocity is

20

m/s. Find the distance (in metres) the particle travels until its velocity falls to

5

m/s. Give your answer to two decimal places.

Q36Straightforward

A particle moves horizontally. Its velocity satisfies

\dfrac{dv}{dt} = -0.5v

with

v(0) = 10

m/s. Find the velocity (in m/s) at

t = 4

s. Give your answer to two decimal places.

Q37Challenging

A particle of mass

1

kg falls with

\dot{v} = 10 - 0.4v^2

(downward positive,

g = 10

m/s²). Find the time (in seconds) to reach half the terminal velocity. Give your answer to two decimal places.

Q38Challenging

A particle of mass

m

falls from rest under gravity with resistance

R = mkv

(proportional to velocity,

k > 0

). Prove that

v = \dfrac{g}{k}\left(1 - e^{-kt}\right)

and state the terminal velocity.

Worked solutions and answers at openmath.au/year-12/extension-2/mechanics