91Ó°ÊÓ

The net force acting on the object can be represented as the difference between the buoyancy force and the weight, minus the water resistance force. The equation for Newton's second law is: \[F_net = ma\] Where F_net is the net force, m is the mass of the object, and a is its acceleration. We can rewrite this equation in terms of the forces on the object: \[F_B - F_W - F_R = ma\] Where F_B is the buoyancy force, F_W is the weight of the object, and F_R is the water resistance force. Given that the buoyancy of the object is 30 lb and the weight is 12 lb, we can rewrite the equation as: \[30 - 12 - F_R = ma\]

As it's given that the water resistance (in pounds) is numerically equal to the square of the velocity (in feet per second), we can represent this as: \[F_R = v^2\] Substitute the water resistance in terms of velocity into the equation from Step 1: \[30 - 12 - v^2 = ma\]

We know the object reaches the surface in 5 seconds, so we need to find the acceleration to determine the velocity at the surface. Based on the equation: \[d = v_0t + \frac{1}{2}at^2\] Where \(d\) is the distance the object travels, \(v_0\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time, we have: \[d = \frac{1}{2}at^2\] Since the object's initial velocity is zero, the initial velocity term (\(v_0t\)) is equal to zero. We will use the given time of 5 seconds to find the acceleration. \[d = \frac{1}{2}a(5)^2\] At the surface, \(d = 0\). So, \[0 = \frac{1}{2}a(5)^2\] Then, we have: \[a = \frac{-2\cdot 0}{(5)^2} = 0\] But this doesn't seem to be right, as the object reaches the surface which means there should be acceleration. So, let's rewrite the Newton's second law equation to be consistent with what's given.

Since the object is moving upward (rising), we need to rewrite the Newton's second law equation to represent the forces acting in the proper direction: \[F_B - F_W - F_R = ma\] \[30 - 12 - v^2 = -ma\] Now, we have: \[18 - v^2 = -ma\]

To find the velocity at the surface, we need to solve the equation for velocity: \[18 - v^2 = -ma\] The negative sign in the equation is actually a result of our coordinate system, in which the direction of the force is negative while the acceleration is positive towards the surface. Let's remove this negative sign as the equation is consistent with the scenario on its own. \[18 - v^2 = ma\] To find the velocity, we can use the kinematic equation: \[v^2 = v_0^2 + 2ad\] At the surface, \(v_0 = 0\) and \(d = 0\), so the equation simplifies to: \[v^2 = 2ad\] Now we can substitute acceleration and distance into the equation: \[v^2 = 2(ma)d\] From Newton's second law equation in Step 4: \[v^2 = 18\] Take the square root of both sides: \[v = \sqrt{18} \approx 4.24 \text{ ft/s}\] So, the velocity of the object at the instant when it reaches the surface is approximately 4.24 feet per second.