Mechanics, EXT2 M1 2022 HSC 8 MC*

As a projectile of mass \(m\) kilograms travels through air, it experiences a frictional force. The magnitude of this force is proportional to 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 \(\displaystyle \binom{x}{y}\). 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?

\(\displaystyle\binom{0}{-m g}+k\binom{\dot{x}}{\dot{y}}=m\binom{\ddot{x}}{\ddot{y}}\)
\(\displaystyle\binom{0}{-m g}-k\binom{\dot{x}}{\dot{y}}=m\binom{\ddot{x}}{\ddot{y}}\)
\(\displaystyle\binom{0}{-m g}+k v\binom{\dot{x}}{\dot{y}}=m\binom{\ddot{x}}{\ddot{y}}\)
\(\displaystyle\binom{0}{-m g}-k v\binom{\dot{x}}{\dot{y}}=m\binom{\ddot{x}}{\ddot{y}}\)

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\(B\)

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\(\text{Friction} \ (F) \ \text{works against velocity}\)

\(\Rightarrow \ \text{the resistance term is subtracted (eliminate A and C)}\)

\(\text{Force vector has magnitude and direction.}\)

\(\text{Magnitude:} \ \ \abs{F} \propto v \ \Rightarrow \ \abs{F}=kv\ \ \ (k>0)\)

\(\text{Direction is in line with unit vector:} \ \ \dfrac{(\dot{x}, \dot{y})}{\abs{(\dot{x}, \dot{y})}}=\dfrac{(\dot{x}, \dot{y})}{v}\)

\(\text{Velocity vector} \ (\dot{x}, \dot{y}) \ \text{has magnitude equal to the speed:}\)

\(\abs{(\dot{x}, \dot{y})}=\sqrt{(\dot{x})^2+(\dot{y})^2}=v\)

\(\abs{-k(\dot{x}, \dot{y})}=k\abs{\dot{x}, \dot{y}}=k v\)

\(\text{Magnitude of resistance} \propto \text{speed}\ \ \Rightarrow\ \ -k \displaystyle\binom{\dot{x}}{\dot{y}}\ \text{(no extra}\ v \ \text{needed)}\)

\(\Rightarrow B\)