Find the velocity of a particle that has the given acceleration and the given initial velocity.
Select the correct answer.
$$\mathbf { a } ( t ) = 3 k , \mathbf { v } ( 0 ) = 16 \mathbf { i } - 3 \mathbf { j }$$
A) $\quad \mathrm { v } ( t ) = 16 \mathbf { i } - \mathbf { j } + 3 t \mathbf { k }$
B) $\mathbf { v } ( t ) = \mathbf { i } - 3 \mathbf { j } + 3 t \mathbf { k }$
C) $\mathbf { v } ( t ) = 16 \mathbf { i } - 3 \mathbf { j } + t \mathbf { k }$
D) $\mathbf { v } ( t ) = 16 \mathbf { i } - 3 \mathbf { j } + 3 t \mathbf { k }$
E) $\mathbf { v } ( t ) = ( 16 t - 9 ) \mathbf { i } + 3 t \mathbf { k }$

Question

Quizplus · Accepted Answer

We know that acceleration is the derivative of velocity with respect to time. So, integrating acceleration gives us velocity. Integrating $\mathbf { a } ( t ) = 3 k$ with respect to time gives us $\mathbf { v } ( t ) = 3 t \mathbf{k} + c$. Now, we need to find the value of constant c using the initial velocity given. $$\mathbf { v } ( 0 ) = 16 \mathbf { i } - 3 \mathbf { j }.$$ Putting t=0 in $\mathbf { v } ( t ) = 3 t \mathbf{k} + c$, we get $\mathbf { v } ( 0 ) = c$. So, c is equal to the initial velocity which is equal to 16$\mathbf { i } - 3\mathbf { j }$. Therefore, Option D is correct, which is $\mathbf { v } ( t ) = 16 \mathbf { i } - 3 \mathbf { j } + 3 t \mathbf { k }$.

Find the Velocity of a Particle That Has the Given a(t)=3k,v(0)=16i−3j\mathbf { a } ( t ) = 3 k , \mathbf { v } ( 0 ) = 16 \mathbf { i } - 3 \mathbf { j }a(t)=3k,v(0)=16i−3j

Find the Velocity of a Particle That Has the Given $\mathbf { a } ( t ) = 3 k , \mathbf { v } ( 0 ) = 16 \mathbf { i } - 3 \mathbf { j }$