Math Snap

STEP 1

1. The box has a mass of $4.0 \, \text{kg}$.
2. The box starts from rest.
3. The box is accelerated at a rate of $2.0 \, \text{m/s}^2$.
4. The acceleration occurs over a time period of $8.0 \, \text{s}$.
5. We need to find the net work done on the box.

STEP 2

1. Calculate the final velocity of the box.
2. Determine the displacement of the box.
3. Calculate the net work done using the work-energy principle.

STEP 3

Calculate the final velocity of the box using the formula for acceleration:
$$v = u + at$$ where $u = 0 \, \text{m/s}$ (initial velocity), $a = 2.0 \, \text{m/s}^2$ (acceleration), and $t = 8.0 \, \text{s}$ (time).
$$v = 0 + (2.0 \, \text{m/s}^2)(8.0 \, \text{s})$$ $$v = 16.0 \, \text{m/s}$$

STEP 4

Determine the displacement of the box using the formula:
$$s = ut + \frac{1}{2}at^2$$ Substitute $u = 0 \, \text{m/s}$, $a = 2.0 \, \text{m/s}^2$, and $t = 8.0 \, \text{s}$:
$$s = 0 + \frac{1}{2}(2.0 \, \text{m/s}^2)(8.0 \, \text{s})^2$$ $$s = \frac{1}{2}(2.0)(64.0)$$ $$s = 64.0 \, \text{m}$$

Calculate the net work done using the work-energy principle:
$$W_{\text{net}} = \Delta KE = \frac{1}{2}mv^2 - \frac{1}{2}mu^2$$ where $m = 4.0 \, \text{kg}$, $v = 16.0 \, \text{m/s}$, and $u = 0 \, \text{m/s}$.
$$W_{\text{net}} = \frac{1}{2}(4.0 \, \text{kg})(16.0 \, \text{m/s})^2 - \frac{1}{2}(4.0 \, \text{kg})(0)^2$$ $$W_{\text{net}} = \frac{1}{2}(4.0)(256.0)$$ $$W_{\text{net}} = 512.0 \, \text{J}$$ The net work done on the box is:
$\boxed{512.0 \, \text{J}}$