To determine if the speed of the particle is increasing, decreasing, or staying the same at $t = 5$ seconds, we need to examine the particle's velocity and acceleration.

Step 1: Find the velocity function.

The velocity of the particle is the first derivative of the position function $x(t)$:

$x(t) = t - 2t^2$

Taking the derivative with respect to time $t$:

$v(t) = \frac{dx(t)}{dt} = 1 - 4t$

So, the velocity function is:

$v(t) = 1 - 4t$

Step 2: Find the acceleration function.

The acceleration is the derivative of the velocity function:

$a(t) = \frac{dv(t)}{dt} = -4$

Since the acceleration is constant at $a(t) = -4$ m/s², the particle is experiencing constant negative acceleration.

Step 3: Evaluate the velocity and acceleration at $t = 5$ seconds.

First, calculate the velocity at $t = 5$:

$v(5) = 1 - 4(5) = 1 - 20 = -19 \, \text{m/s}$

The velocity is negative, indicating that the particle is moving in the negative direction.

Now, calculate the acceleration at $t = 5$:

$a(5) = -4 \, \text{m/s}^2$

Step 4: Determine if the speed is increasing or decreasing.

Speed is the magnitude of velocity. Since the velocity is negative ($v(5) = -19$), the speed at $t = 5$ seconds is 19 m/s. The key is to examine whether the speed is increasing or decreasing by considering the signs of the velocity and acceleration.

• The particle is moving in the negative direction (since $v(5) = -19$).
• The acceleration is negative ($a(t) = -4$), which means the particle is slowing down in the negative direction (decreasing the magnitude of its velocity).

Therefore, the speed is decreasing at $t = 5$ seconds.

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Would you like further details or have any questions about the process?

Here are 5 related questions you might find interesting:

1. How do you interpret the sign of velocity and acceleration to determine speed?
2. How can you determine the turning points in the motion of a particle from its position function?
3. How do you find when the particle momentarily stops (velocity equals zero)?
4. What would happen if the acceleration were positive instead of negative?
5. Can you determine the particle's total distance traveled from its position function?

Tip: When analyzing motion, always consider both the velocity (direction) and acceleration (change in velocity) to determine if speed is increasing or decreasing.