Use the given acceleration function and initial conditions to find the position at time t = 1. $$\mathbf { a } ( t ) = 10 \mathbf { i } + 6 \mathbf { j } + 8 \mathbf { k } , \quad \mathbf { v } ( 0 ) = \mathbf { 0 } , \quad \mathbf { r } ( 0 ) = 5 \mathbf { j }$$
A) $\mathbf { r } ( 1 ) = 11 \mathbf { i } + 8 \mathbf { j } + 4 \mathbf { k }$
B) $\mathbf { r } ( 1 ) = 5 \mathbf { i } + 4 \mathbf { j } - 4 \mathbf { k }$
C) $\mathbf { r } ( 1 ) = 5 \mathbf { i } + 8 \mathbf { j } + 4 \mathbf { k }$
D) $\mathbf { r } ( 1 ) = 5 \mathbf { i } + 8 \mathbf { j } - 4 \mathbf { k }$
E) $\mathbf { r } ( 1 ) = 5 \mathbf { i } + 8 \mathbf { j } + 9 \mathbf { k }$

Question

Quizplus · Accepted Answer

Using the equations of motion, we can find the velocity and position functions:
$$\mathbf { v } ( t ) = \int \mathbf { a } ( t ) dt = 10t \mathbf { i } + 6t \mathbf { j } + 8t \mathbf { k } + \mathbf { v } ( 0 ) = 10t \mathbf { i } + 6t \mathbf { j } + 8t \mathbf { k }$$
$$\mathbf { r } ( t ) = \int \mathbf { v } ( t ) dt = 5\mathbf { i } + 3t^2 \mathbf { j } + 4t^2 \mathbf { k } + \mathbf { r } ( 0 ) = 5 \mathbf { i } + 3t^2 \mathbf { j } + 4t^2 \mathbf { k }$$
Plugging in t=1, we get:
$$\mathbf { r } ( 1 ) = 5 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } = 5 \mathbf { i } + 8 \mathbf { j } + 4 \mathbf { k }$$
Therefore, the best choice is C.

Use the Given Acceleration Function and Initial Conditions to Find