Determine the kinetic energy of the object as a function of time.

The rate at which the resultant force is doing work is defined as power (P). Power can be calculated as the time derivative of kinetic energy, or as the dot product of force and velocity (P = F · v). Step 1: Determine the kinetic energy of the object as a function of time. The mass of the object is m = 4.0 kg. The speed of the object is given by v = (2t) m/s. The kinetic energy (K) is given by the formula K = (1)/(2)mv^2. Substitute the given values: K = (1)/(2)(4.0 \, kg)(2t)^2 K = (1)/(2)(4.0 \, kg)(4t^2 \, m^2/s^2) K = (2.0)(4t^2) \, J K = 8.0t^2 \, J Step 2: Calculate the power by differentiating the kinetic energy with respect to time. Power (P) is the rate of change of kinetic energy: P = (dK)/(dt). P = (d)/(dt)(8.0t^2) P = 8.0 × 2t P = 16.0t \, W Step 3: Evaluate the power at the specified time t = 1 s. Substitute t = 1 s into the power equation: P = 16.0(1) \, W P = 16 \, W Alternatively, using P = Fv: Step 1: Calculate the acceleration of the object. Given v = 2t, the acceleration a = (dv)/(dt). a = (d)/(dt)(2t) = 2 \, m/s^2 Step 2: Calculate the resultant force on the object. Using Newton's second law, F = ma. F = (4.0 \, kg)(2 \, m/s^2) F = 8.0 \, N Step 3: Calculate the speed of the object at t = 1 s. v(1) = 2(1) = 2 \, m/s Step 4: Calculate the power. Since the force causes the acceleration in the direction of motion, the force and velocity are in the same direction. P = Fv P = (8.0 \, N)(2 \, m/s) P = 16 \, W Both methods yield the same result. The final answer is b. 16 W. That's 2 down. 3 left today — send the next one.