Certainly! Let's solve this step-by-step based on the given problem.
### Problem Statement
The motion of a particle in a plane is defined by the equations:
[tex]$ x(t) = 13 + 3t - 4 $[/tex]
[tex]$ y(t) = 12 - 2t + 6 $[/tex]
We need to determine the velocity and acceleration of the particle after 3 seconds (t = 3 seconds).
### Step 1: Calculate the Position at t = 3 seconds
We need to determine the position of the particle at [tex]\( t = 3 \)[/tex] seconds using the given equations for [tex]\( x(t) \)[/tex] and [tex]\( y(t) \)[/tex].
#### For x(t):
[tex]$ x(3) = 13 + 3(3) - 4 $[/tex]
[tex]\[
= 13 + 9 - 4 \\
= 18
\][/tex]
#### For y(t):
[tex]$ y(3) = 12 - 2(3) + 6 $[/tex]
[tex]\[
= 12 - 6 + 6 \\
= 12
\][/tex]
So, the position of the particle at [tex]\( t = 3 \)[/tex] seconds is [tex]\( (18, 12) \)[/tex] meters.
### Step 2: Determine the Velocity
Velocity is the first derivative of the position with respect to time.
#### For x(t):
The velocity in the x-direction, [tex]\( v_x \)[/tex], is the derivative of [tex]\( x(t) \)[/tex]:
[tex]$ \frac{dx}{dt} = \frac{d}{dt}(13 + 3t - 4) \\
= 3
$[/tex]
#### For y(t):
The velocity in the y-direction, [tex]\( v_y \)[/tex], is the derivative of [tex]\( y(t) \)[/tex]:
[tex]$ \frac{dy}{dt} = \frac{d}{dt}(12 - 2t + 6) \\
= -2
$[/tex]
Therefore, the velocity of the particle at [tex]\( t = 3 \)[/tex] seconds is [tex]\( (3, -2) \)[/tex] meters per second.
### Step 3: Determine the Acceleration
Acceleration is the second derivative of the position with respect to time or the first derivative of velocity with respect to time.
#### For x(t):
The acceleration in the x-direction, [tex]\( a_x \)[/tex], is the derivative of [tex]\( v_x \)[/tex]:
[tex]$ \frac{d^2x}{dt^2} = \frac{d}{dt}(3) \\
= 0
$[/tex]
#### For y(t):
The acceleration in the y-direction, [tex]\( a_y \)[/tex], is the derivative of [tex]\( v_y \)[/tex]:
[tex]$ \frac{d^2y}{dt^2} = \frac{d}{dt}(-2) \\
= 0
$[/tex]
Therefore, the acceleration of the particle at [tex]\( t = 3 \)[/tex] seconds is [tex]\( (0, 0) \)[/tex] meters per second squared.
### Summary
- Position at [tex]\( t = 3 \)[/tex] seconds: [tex]\( (18, 12) \)[/tex] meters
- Velocity at [tex]\( t = 3 \)[/tex] seconds: [tex]\( (3, -2) \)[/tex] meters per second
- Acceleration at [tex]\( t = 3 \)[/tex] seconds: [tex]\( (0, 0) \)[/tex] meters per second squared
