Math Problem Statement

Find the position vector of a particle that has the given acceleration and the specified initial velocity and position. a(t) = 17t i + et j + e−t k, v(0) = k, r(0) = j + k

Solution

To find the position vector r(t)\mathbf{r}(t) of the particle, we need to integrate the given acceleration vector a(t)\mathbf{a}(t) twice and apply the initial conditions for velocity and position.

Step 1: Find the velocity vector v(t)\mathbf{v}(t)

The acceleration vector is given as: a(t)=17ti^+etj^+etk^\mathbf{a}(t) = 17t\,\hat{i} + e^t\,\hat{j} + e^{-t}\,\hat{k} We integrate a(t)\mathbf{a}(t) to find the velocity vector v(t)\mathbf{v}(t).

Velocity in the i^\hat{i}-direction:

vi(t)=17tdt=17t22+Civ_i(t) = \int 17t \, dt = \frac{17t^2}{2} + C_i

Velocity in the j^\hat{j}-direction:

vj(t)=etdt=et+Cjv_j(t) = \int e^t \, dt = e^t + C_j

Velocity in the k^\hat{k}-direction:

vk(t)=etdt=et+Ckv_k(t) = \int e^{-t} \, dt = -e^{-t} + C_k

Thus, the general form of the velocity vector is: v(t)=(17t22+Ci)i^+(et+Cj)j^+(et+Ck)k^\mathbf{v}(t) = \left( \frac{17t^2}{2} + C_i \right)\hat{i} + (e^t + C_j)\hat{j} + \left( -e^{-t} + C_k \right)\hat{k}

Step 2: Apply the initial condition v(0)=k^\mathbf{v}(0) = \hat{k}

From v(0)=k^\mathbf{v}(0) = \hat{k}, we know: v(0)=Cii^+(e0+Cj)j^+(e0+Ck)k^=0i^+0j^+1k^\mathbf{v}(0) = C_i\,\hat{i} + (e^0 + C_j)\,\hat{j} + \left( -e^0 + C_k \right)\hat{k} = 0\,\hat{i} + 0\,\hat{j} + 1\,\hat{k}

This gives the following conditions: Ci=0,e0+Cj=0    Cj=1,e0+Ck=1    Ck=2C_i = 0, \quad e^0 + C_j = 0 \implies C_j = -1, \quad -e^0 + C_k = 1 \implies C_k = 2

Thus, the velocity vector becomes: v(t)=17t22i^+(et1)j^+(et+2)k^\mathbf{v}(t) = \frac{17t^2}{2}\,\hat{i} + (e^t - 1)\,\hat{j} + (-e^{-t} + 2)\,\hat{k}

Step 3: Find the position vector r(t)\mathbf{r}(t)

We now integrate v(t)\mathbf{v}(t) to find r(t)\mathbf{r}(t).

Position in the i^\hat{i}-direction:

ri(t)=17t22dt=17t36+Dir_i(t) = \int \frac{17t^2}{2} \, dt = \frac{17t^3}{6} + D_i

Position in the j^\hat{j}-direction:

rj(t)=(et1)dt=ett+Djr_j(t) = \int (e^t - 1) \, dt = e^t - t + D_j

Position in the k^\hat{k}-direction:

rk(t)=(et+2)dt=et+2t+Dkr_k(t) = \int (-e^{-t} + 2) \, dt = e^{-t} + 2t + D_k

Thus, the general form of the position vector is: r(t)=(17t36+Di)i^+(ett+Dj)j^+(et+2t+Dk)k^\mathbf{r}(t) = \left( \frac{17t^3}{6} + D_i \right)\hat{i} + (e^t - t + D_j)\hat{j} + \left( e^{-t} + 2t + D_k \right)\hat{k}

Step 4: Apply the initial condition r(0)=j^+k^\mathbf{r}(0) = \hat{j} + \hat{k}

From r(0)=j^+k^\mathbf{r}(0) = \hat{j} + \hat{k}, we know: r(0)=Dii^+(e00+Dj)j^+(e0+0+Dk)k^=0i^+1j^+1k^\mathbf{r}(0) = D_i\,\hat{i} + (e^0 - 0 + D_j)\,\hat{j} + \left( e^0 + 0 + D_k \right)\,\hat{k} = 0\,\hat{i} + 1\,\hat{j} + 1\,\hat{k}

This gives the following conditions: Di=0,e0+Dj=1    Dj=0,e0+Dk=1    Dk=0D_i = 0, \quad e^0 + D_j = 1 \implies D_j = 0, \quad e^0 + D_k = 1 \implies D_k = 0

Thus, the position vector becomes: r(t)=17t36i^+(ett)j^+(et+2t)k^\mathbf{r}(t) = \frac{17t^3}{6}\,\hat{i} + (e^t - t)\,\hat{j} + (e^{-t} + 2t)\,\hat{k}

Final Answer:

The position vector of the particle is: r(t)=17t36i^+(ett)j^+(et+2t)k^\mathbf{r}(t) = \frac{17t^3}{6}\,\hat{i} + (e^t - t)\,\hat{j} + (e^{-t} + 2t)\,\hat{k}

Would you like further details on any part of the solution or have any questions?


Here are 5 related questions:

  1. How do you find velocity from a given acceleration vector?
  2. What is the relationship between position, velocity, and acceleration vectors?
  3. Can you derive the position vector if given initial conditions for position only?
  4. How does the exponential function behave in differential equations?
  5. What is the physical interpretation of the constants found during integration?

Tip: Always apply initial conditions immediately after integration to simplify the process of finding constants.

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Math Problem Analysis

Mathematical Concepts

Calculus
Vector Calculus
Differential Equations
Kinematics

Formulas

v(t) = ∫a(t) dt
r(t) = ∫v(t) dt

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Undergraduate (Calculus)