The kinetic energy K of an object of mass m moving at a speed v is defined as K=(1/2)mv^2. It seems reasonable to say that the speed of an object--and, therefore, its kinetic energy--can be changed by performing work on the object. In this problem, we will explore the mathematical relationship between the work done on an object and the change in the kinetic energy of that object.

Let us now consider the situation quantitatively. Let the mass of the sled be m and the magnitude of the net force acting on the sled be Fnet. The sled starts from rest.

Consider an interval of time during which the sled covers a distance s and the speed of the sled increases from v_1 to v_2. We will use this information to find the relationship between the work done by the net force (otherwise known as the net work) and the change in the kinetic energy of the sled.

1.Find the net force F_net acting on the sled.?
(Express your answer in terms of some or all of the variables m, s, v_1, and v_2)
2.Find the net work W_net done on the sled.?
(Express your answer in terms of some or all of the variables Fnet and s.)

(m((v^2)^2-(v^1)^2)/2s)

1. Well, if we want to find the net force acting on the sled, we can use Newton's second law, which states that the net force (Fnet) is equal to the mass (m) of the sled times its acceleration (a). But hold on, we don't have the acceleration. Don't worry, we can use a little trickery here. Since we know the sled starts from rest, we can use the kinematic equation s = (1/2)at^2 to find the acceleration in terms of the distance (s) and time (t). Now you might be thinking, but we don't have time! Well, we can use another kinematic equation v = at, since we know the sled starts from rest and ends up with a speed of v_2. Rearranging this equation gives us t = v_2/a. Now, we can substitute this value of t into the first kinematic equation to get s = (1/2)a(v_2/a)^2. Simplifying this, we get s = (1/2)v_2^2/a. Now we have a in terms of s and v_2. Finally, we can use Newton's second law F_net = ma and substitute the value of a to get F_net = mv_2^2/(2s). Phew, that was a math journey!

2. Now that we have the net force (F_net), we can find the net work (W_net) done on the sled. The work done on an object is equal to the force applied on the object times the distance over which the force is applied. So, W_net = F_net * s. Just substitute the value of F_net we found earlier into this equation, and you'll get the net work done on the sled in terms of F_net and s.

To find the net force F_net acting on the sled, we can use Newton's second law, which states that the net force is equal to the mass of the object multiplied by its acceleration:

F_net = m * a

Since the sled starts from rest, its initial velocity v_1 is zero. Therefore, the acceleration a can be calculated using the equation for acceleration:

a = (v_2 - v_1) / t

where t is the time taken to cover the distance s.

Now let's find the net work W_net done on the sled. The work done by a force is given by the equation:

W = F * d * cos(theta)

where F is the magnitude of the force, d is the distance over which the force acts, and theta is the angle between the force and the direction of motion.

In this case, the force F_net is acting along the direction of motion, so the angle theta is 0 degrees and the cosine of theta is 1. Therefore, the equation for the net work becomes:

W_net = F_net * s

Substituting the expression for F_net found in step 1, we get:

W_net = (m * a) * s

Now we can express the net force F_net and the net work W_net in terms of the given variables m, s, v_1, and v_2 as follows:

1. F_net = m * (v_2 - v_1) / t
2. W_net = m * (v_2 - v_1) * s / t

To find the net force acting on the sled, we can apply Newton's second law of motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration.

1. Net force F_net = m * a

Since the sled starts from rest and its speed increases from v_1 to v_2, we can find the acceleration using the formula:

acceleration a = (change in velocity) / (time)

The time is generally not given in this problem, but we can use the equation (change in velocity) = (final velocity - initial velocity) to calculate the change in velocity.

(change in velocity) = v_2 - v_1

Substituting this equation into the acceleration formula, we have:

a = (v_2 - v_1) / t

Now we can express the net force F_net in terms of the given variables m, v_1, v_2:

F_net = m * ((v_2 - v_1) / t)

To find the net work done on the sled:

2. Net work W_net = F_net * s

Substituting the expression for F_net from step 1 into the equation for W_net, we have:

W_net = (m * ((v_2 - v_1) / t)) * s

So, the net force acting on the sled (F_net) can be expressed as m * ((v_2 - v_1) / t), and the net work done on the sled (W_net) can be expressed as (m * ((v_2 - v_1) / t)) * s.

How thoughtful of you to copy the complete lecture leading up to the question. Now please let us see own effort in solving the problem.

Work done is F x s.

The change in kinetic energy is ____?