Displacement, Velocity and Distance Travelled by Natural Logarithmic Equations

A particle is moving in a straight line, starting from the origin. At time \( t \) seconds the particle has a displacement of \( x \) metres from the origin and a velocity \( v \ \text{m s}^{-1} \).
The displacement is given by \( x = 3t – 6 \log_e (t+1) \).

Part 1

Find the expression for the velocity, \( v \).

\( \begin{align} \displaystyle v &= \frac{dx}{dt} \\ &= \frac{d}{dx} 3t – 6 \frac{d}{dx} \log_e (t+1) \\ &= 3 – 6 \times \frac{1}{t+1} \\ &= 3 – \frac{6}{t+1} \end{align} \)

Part 2

Find the initial velocity.

\( \begin{align} \displaystyle v (t=0) &= 3 – \frac{6}{0+1} \\ &= -3 \end{align} \)

Part 3

Determine the initial direction of the particle.

Moving backward as \( v = -3 \lt 0 \)

Part 4

Find when the particle comes to rest.

Particle comes to rest when \( v=0 \)
\( \displaystyle \begin{align} 3 – \frac{6}{t+1} &= 0 \\ -\frac{6}{t+1} &= -3 \\ \frac{6}{t+1} &= 3 \\ t+1 &= 2 \\ t &= 1 \end{align} \)

Part 5

Find the distance travelled for the first second.

\( \displaystyle \begin{align} x(t=0) &= 0 \\ x(t=1) &= 3 \times 1 – 6 \log_e (1+1) \\ & = 3 – 6 \log_e 2 \lt 0 \\ \text{distance travelled} &= 6 \log_e 2 -3 &\color{green}{\cdots \text{moving backward} } \end{align} \)

Part 6

Find the distance travelled that the particle comes back to the origin.

\( 2 \times (6 \log_e 2 – 3) = 12 \log_e 2 – 6 \)

Part 7

Find the displacement at \( t = 4 \).

\( \begin{align} \displaystyle x(t=4) &= 3 \times 4 – 6 \log_e (4+1) \\ &= 12 – 6 \log_e 5 \end{align} \)

Part 8

Find the distance travelled by the particle in the first four seconds.

\( \begin{align} (12 \log_e 2 – 6) + (12 – 6 \log_e 5) &= 6 + 12 \log_e 2 + 6 \log_e 5 \\ &= 23.974 \cdots \text{metres} \end{align} \)

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorials Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume

 



Your email address will not be published. Required fields are marked *