While an elevator of mass $908 \mathrm{~kg}$ moves downward, the tension in the supporting cable is a constant $7730 \mathrm{~N}$. Between $t=0$ and $t=$ $4.00 \mathrm{~s}$, the elevator's displacement is $5.00 \mathrm{~m}$ downward. What is the elevator's speed at $t=4.00 \mathrm{~s} ?$
$\mathrm{m} / \mathrm{s}$

Step 1 ：Given that the mass of the elevator is \(908 \, \mathrm{kg}\), the tension in the supporting cable is \(7730 \, \mathrm{N}\), and the displacement of the elevator is \(5.00 \, \mathrm{m}\) downward between \(t=0\) and \(t=4.00 \, \mathrm{s}\).

Step 2 ：The net force acting on the elevator is the difference between the gravitational force and the tension in the cable. The gravitational force can be calculated as \(908 \, \mathrm{kg} \times 9.8 \, \mathrm{m/s^2} = 8898.4 \, \mathrm{N}\). Therefore, the net force is \(8898.4 \, \mathrm{N} - 7730 \, \mathrm{N} = 1168.4 \, \mathrm{N}\).

Step 3 ：The acceleration of the elevator can be found by dividing the net force by the mass of the elevator. Therefore, the acceleration is \(1168.4 \, \mathrm{N} / 908 \, \mathrm{kg} = 1.29 \, \mathrm{m/s^2}\).

Step 4 ：The speed of the elevator at \(t=4.00 \, \mathrm{s}\) can be found by multiplying the acceleration by the time. Therefore, the speed is \(1.29 \, \mathrm{m/s^2} \times 4.00 \, \mathrm{s} = 5.15 \, \mathrm{m/s}\).

Step 5 ：\(\boxed{5.15 \, \mathrm{m/s}}\) is the elevator's speed at \(t=4.00 \, \mathrm{s}\).