Question

A boater and motor boat are at rest on a lake. Together, they have mass 200.0 kg. If the thrust of the motor is a constant force of 40.0 N in the direction of motion, and if the resistive force of the water is numerically equivalent to 2 times the speed v of the boat, set up and solve the differential equation to find:

(a) the velocity of the boat at time t;

(a) 4 m/s
(b) 2 m/s
(c) 1 m/s
(d) 3 m/s

Answer 1

To find the velocity of the boat at time t, we set up and solve the differential equation 40 - 2v = 200a. However, without additional information about the initial conditions, we cannot determine the exact values of v and a at any given time t. The limiting velocity of the boat is found by setting a = 0 in the equation, which gives v = 20 m/s.

Step-by-step explanation:

To set up the differential equation, we start with Newton's second law: F = ma, where F is the net force acting on the boat, m is the mass of the boat, and a is the acceleration of the boat.

The net force is the difference between the thrust force and the resistive force: F = 40 - 2v, where v is the velocity of the boat.

Plugging in the values, we get: 40 - 2v = ma.

Since the mass of the boat is given as 200.0 kg, the differential equation becomes: 40 - 2v = 200a.

To solve for v, we can rearrange the equation as: 2v + 200a = 40.

Now, we can use calculus to solve this differential equation. However, without additional information about the initial conditions, it is not possible to determine the exact values of v and a at any given time t. Therefore, we cannot determine the velocity of the boat at time t. However, we can solve for the limiting velocity by setting a = 0. From the equation 2v + 200(0) = 40, we find that the limiting velocity is v = 20 m/s.