3D Trajectory Analysis Using Vector Parametrics

00:00 10 marks

Consider a particle moving in 3D space with position vector \(\mathbf{r}(t) = (3t + 2, t^2 – 1, 4 – t)\) where \(t\) is the time in seconds.

(a) Express the motion in terms of its directional components and determine the velocity vector \(\mathbf{v}(t)\).

(b) Determine if there exists a time \(t\) when the particle collides with an object located at fixed position \((11, 8, 2)\), and if so, find the time of collision.

Part a

6 minutes 5 marks

Question

Express the motion by writing the position functions for each coordinate and differentiate them to obtain the velocity vector \(\mathbf{v}(t)\). Clearly state the derivatives for the x, y, and z components.

Hint Hint

Hint

Differentiate each component of \(\mathbf{r}(t)\) with respect to \(t\) using basic differentiation rules.

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Complete Answer

The position functions are \(x(t) = 3t+2\), \(y(t) = t^2-1\), \(z(t) = 4-t\).

Differentiating each component with respect to \(t\) gives:
\(v_x(t) = \frac{dx}{dt} = 3\)
\(v_y(t) = \frac{dy}{dt} = 2t\)
\(v_z(t) = \frac{dz}{dt} = -1\)

Thus, the velocity vector is \(\mathbf{v}(t) = (3, 2t, -1)\).

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Part b

6 minutes 5 marks

Question

Determine if there exists a time \(t\) when \(\mathbf{r}(t)\) equals the position \((11, 8, 2)\). If such a time exists, find \(t\), providing justification for each coordinate equality.

Hint Hint

Hint

To check for collision, set each component of \(\mathbf{r}(t)\) equal to the corresponding component of \((11, 8, 2)\) and solve the resulting system of equations for \(t\). A collision occurs only if there is a common solution for \(t\) in all three equations.

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Complete Answer

To check for collision, we set \(\mathbf{r}(t) = (11, 8, 2)\), which gives us the following system of equations:
\(3t+2 = 11\)
\(t^2-1 = 8\)
\(4-t = 2\)

Solving the first equation: \(3t+2 = 11 \Rightarrow 3t = 9 \Rightarrow t = 3\).

Solving the third equation: \(4-t = 2 \Rightarrow t = 2\).

Solving the second equation: \(t^2-1 = 8 \Rightarrow t^2 = 9 \Rightarrow t = \pm 3\).

For a collision to occur, there must be a common value of \(t\) that satisfies all three equations. We see that from the x-component equation, \(t=3\), and from the z-component equation, \(t=2\). These values are inconsistent, meaning there is no single time \(t\) when the particle is at position \((11, 8, 2)\). Therefore, no collision occurs.

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