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Mechanics, EXT2 M1 2025 HSC 16b

A particle of mass 1 kg is projected from the origin with a speed of 50 ms\(^{-1}\), at an angle of \(\theta\) below the horizontal into a resistive medium.
 

The position of the particle \(t\) seconds after projection is \((x, y)\), and the velocity of the particle at that time is  \(\underset{\sim}{v}=\displaystyle \binom{\dot{x}}{\dot{y}}\).

The resistive force, \(\underset{\sim}{R}\), is proportional to the velocity of the particle, so that  \(\underset{\sim}{R}=-k \underset{\sim}{v}\), where \(k\) is a positive constant.

Taking the acceleration due to gravity to be 10 ms\(^{-2}\), and the upwards vertical direction to be positive, the acceleration of the particle at time \(t\) is given by:

\(\underset{\sim}{a}=\displaystyle \binom{-k \dot{x}}{-k \dot{y}-10}\).    (Do NOT prove this.) 

Derive the Cartesian equation of the motion of the particle, given  \(\sin \theta=\dfrac{3}{5}\).   (5 marks)

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\(y=\left(\dfrac{1-3 k}{4 k}\right) x+\dfrac{10}{k^2} \times \ln \abs{1-\dfrac{k x}{40}}\)

Show Worked Solution

\(\sin \theta=\dfrac{3}{5} \ \Rightarrow \ \cos \theta=\dfrac{4}{5}\)

\(\text{Components of initial velocity:}\)

\(\dot{x}(0)=50\, \cos \theta=50 \times \dfrac{4}{5}=40 \ \text{ms}^{-1}\)

\(\dot{y}(0)=50\, \sin \theta=50 \times \dfrac{3}{5}=-30\ \text{ms}^{-1}\)

♦♦ Mean mark 35%.

\(\text{Horizontal motion:}\)

  \(\dfrac{d \dot{x}}{dt}\) \(=-k \dot{x} \ \ \text{(given)}\)  
\(\dfrac{dt}{d \dot{x}}\) \(=-\dfrac{1}{k \dot{x}}\)  
\(\displaystyle \int dt\) \(=-\dfrac{1}{k} \int \dfrac{1}{\dot{x}}\, d x\)  
\(t\) \(=-\dfrac{1}{k} \ln \dot{x}+c\)  

 
\(\text{When} \ \ t=0, \ \dot{x}=40 \ \Rightarrow \ c=\dfrac{1}{k} \ln 40\)

\(t\) \(=\dfrac{1}{k} \ln 40-\dfrac{1}{k} \ln \abs{\dot{x}}=\dfrac{1}{k} \ln \abs{\dfrac{40}{\dot{x}}}\)
    \(k t\) \(=\ln \abs{\dfrac{40}{\dot{x}}}\)
  \(e^{k t}\) \(=\dfrac{40}{\dot{x}}\)
\(\dot{x}\) \(=40 e^{-k t}\)
\(x\) \(\displaystyle=\int 40 e^{-k t}\, d t=-\dfrac{40}{k} \times e^{-k t}+c\)

 

\(\text{When} \ \ t=0, x=0 \ \Rightarrow \ c=\dfrac{40}{k}\)

   \(x=\dfrac{40}{k}-\dfrac{40}{k} e^{-k t}=\dfrac{40}{k}\left(1-e^{-k t}\right)\ \ldots\ (1)\)
 

\(\text{Vertical Motion }\)

\(\dfrac{d \dot{y}}{dt}\) \(=-k \dot{y}-10 \quad \text{(given)}\)
\(\dfrac{d t}{d \dot{y}}\) \(=-\dfrac{1}{k} \times \dfrac{1}{\dot{y}+\frac{10}{k}}\)
\(t\) \(=-\dfrac{1}{k} \displaystyle \int \dfrac{1}{\dot{y}+\frac{10}{k}} \, d \dot{y}=-\dfrac{1}{k} \ln \abs{\dot{y}+\frac{10}{k}}+c\)

 

\(\text{When} \ \ t=0, \, \dot{y}=-30 \ \ \Rightarrow\ \ c=\dfrac{1}{k} \ln \abs{-30+\frac{10}{k}}\)

\(t\) \(=\dfrac{1}{k} \ln \abs{-30+\frac{10}{k}}-\dfrac{1}{k} \ln \abs{\dot{y}+\frac{10}{k}}\)
  \(=\dfrac{1}{k} \ln \abs{\frac{-30+\frac{10}{k}}{\dot{y}+\frac{10}{k}}}\)
  \(e^{k t}\) \(=\abs{\dfrac{-30+\frac{10}{k}}{y+\frac{10}{k}}}\)
\(\dot{y}\) \(=\left(-30+\dfrac{10}{k}\right) e^{-k t}-\dfrac{10}{k}\)
\(y\) \(=\displaystyle \left(-30+\dfrac{10}{k}\right) \int e^{-kt}\, d t-\int \dfrac{10}{k}\, dt\)
  \(=-\dfrac{1}{k}\left(-30+\dfrac{10}{k}\right) e^{-k t}-\dfrac{10 t}{k}+c\)

 \(\text{When} \ \ t=0, y=0 \ \Rightarrow \  c=\dfrac{1}{k}\left(-30+\dfrac{10}{k}\right)\)

  \(y\) \(=\dfrac{1}{k}\left(-30+\dfrac{10}{k}\right)-\dfrac{1}{k}\left(-30+\dfrac{10}{k}\right) e^{-k t}-\dfrac{10t}{k}\)
  \(=\left(\dfrac{10}{k^2}-\dfrac{30}{k}\right)\left(1-e^{-k t}\right)-\dfrac{10 t}{k}\ \ldots\ (2)\)

 

\(\text {Cartesian equation (using (1) above):}\)

\(x\) \(=\dfrac{40}{k}\left(1-e^{-k t}\right)\)
\(\dfrac{k x}{40}\) \(=1-e^{-k t}\)
\(e^{-k t}\) \(=1-\dfrac{k x}{40}\)
\(-k t\) \(=\ln \abs{1-\dfrac{k x}{40}}\)
\(t\) \(=-\dfrac{1}{k} \ln \abs{1-\dfrac{kx}{40}}\)

 

\(y\) \(=\left(\dfrac{10}{k^2}-\dfrac{30}{k}\right) \times \dfrac{k x}{40}+\dfrac{10}{k^2} \times \ln \abs{1-\dfrac{k x}{40}}\)
  \(=\left(\dfrac{1-3 k}{4 k}\right) x+\dfrac{10}{k^2} \times \ln \abs{1-\dfrac{k x}{40}}\)