As a projectile of mass $m$ kilograms travels through air, it experiences a frictional force. The magnitude of this force is proportional to the square of the speed $v$ of the projectile. The constant of proportionality is the positive number $k$. The position of the particle at time $t$ is denoted by $\vec{r}(t)$. The acceleration due to gravity is $g \, \text{m s}^{-2}$. Based on Newton's laws of motion, which equation models the motion of this projectile?

SSCE HSC Mathematics Extension 2 Forces and equations of motion: As a projectile of mass $m$ kilo

As a projectile of mass $m$ kilograms travels through air, it experiences a frictional force - HSC - SSCE Mathematics Extension 2 - Question 8 - 2022 - Paper 1

Question 8

As a projectile of mass $m$ kilograms travels through air, it experiences a frictional force. The magnitude of this force is proportional to the square of the speed ... show full transcript

Worked Solution & Example Answer:As a projectile of mass $m$ kilograms travels through air, it experiences a frictional force - HSC - SSCE Mathematics Extension 2 - Question 8 - 2022 - Paper 1

Step 1

Identify the forces acting on the projectile

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Answer

The forces acting on the projectile include:

The gravitational force downwards, given by: $\vec{F}_g = -mg\hat{j}$ , where $\hat{j}$ is the unit vector upward.
The frictional force, which opposes the motion and is given by: $\vec{F}_f = -k v^2 \hat{v}$ , where $\hat{v}$ is the unit vector in the direction of velocity.

Step 2

Apply Newton's second law

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Answer

According to Newton's second law, the net force acting on the projectile is equal to the mass of the projectile multiplied by its acceleration:

$\vec{F}_{net} = m\vec{a}$

Thus, we can express the net force as:

$\vec{F}_{net} = \vec{F}_g + \vec{F}_f = -mg\hat{j} - k v^2 \hat{v}$

Step 3

Combine the forces to form the equation of motion

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Answer

Substituting into Newton's second law gives:

$-mg\hat{j} - k v^2 \hat{v} = m\vec{a}$

This equation captures the motion of the projectile, where the terms on the left represent the forces and the term on the right represents the mass times the acceleration.

Step 4

Choose the correct model from the options

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Answer

The derived equation matches option B:

$(-mg \hat{j} - k v^2 \hat{v} = m \vec{a})$

Hence, the correct answer is B.

As a projectile of mass $m$ kilograms travels through air, it experiences a frictional force - HSC - SSCE Mathematics Extension 2 - Question 8 - 2022 - Paper 1

Worked Solution & Example Answer:As a projectile of mass $m$ kilograms travels through air, it experiences a frictional force - HSC - SSCE Mathematics Extension 2 - Question 8 - 2022 - Paper 1

Identify the forces acting on the projectile

Apply Newton's second law

Combine the forces to form the equation of motion

Choose the correct model from the options

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