Two carts, one twice as heavy as the other, are at rest on a horizontal frictionless track. A person pushes each cart with the same force for 6.00 s.

If the kinetic energy of the lighter cart after the push is K, the kinetic energy of the heavier cart is

A. 1/4 K
B. 1/2 K
C. K
D. 2K
E. 4K

See below

The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. Therefore, the work done on each cart by the person pushing it is the same.

Let's assume the applied force is F and the displacement of each cart is equal, denoted by d.

The work done on each cart is given by the equation:

Work = Force x Displacement

Now, since the force and displacement are the same for both carts, the work done on each cart is equal.

W = F * d

The change in kinetic energy of each cart is given by the equation:

ΔKE = KE_final - KE_initial

Now, let's consider the light cart first. The work done on the light cart is equal to its change in kinetic energy:

W_light = ΔKE_light

Now, let's consider the heavy cart. The work done on the heavy cart is equal to its change in kinetic energy:

W_heavy = ΔKE_heavy

Given that the light cart has a kinetic energy of K after the push, we can write the equation as:

W_light = K

Since the force and displacement are the same for both carts, the work done on the heavy cart is also equal to K:

W_heavy = K

Therefore, the kinetic energy of the heavier cart is K, which corresponds to option C.

To answer this question, let's first understand the concept of kinetic energy and how it relates to the mass and speed of an object.

Kinetic energy (K) is the energy of an object due to its motion. It depends on two factors: the mass (m) of the object and its speed (v). The formula for kinetic energy is K = (1/2)mv^2.

Given that the lighter cart has a kinetic energy of K, it means that the kinetic energy of the lighter cart is given by K = (1/2)m₁v₁^2, where m₁ is the mass of the lighter cart and v₁ is its speed.

Now, let's consider the heavier cart. We are given that it is twice as heavy as the lighter cart. Let's denote the mass of the heavier cart as m₂. Since it is twice as heavy as the lighter cart, we can write m₂ = 2m₁.

Given that both carts are pushed with the same force for the same duration, they experience the same impulse (change in momentum). In other words, the change in momentum of both carts is equal.

Since the mass of the heavier cart is twice as large as the lighter cart, its velocity will change half as much as the lighter cart. This is because the change in momentum is directly proportional to the mass and inversely proportional to the velocity (as per Newton's second law).

However, the kinetic energy depends on the square of the velocity. So, because the velocity of the heavier cart changes half as much as the lighter cart, the change in kinetic energy of the heavier cart will be one-quarter of the change in kinetic energy of the lighter cart.

Therefore, the kinetic energy of the heavier cart after the push is (1/4)K, which corresponds to option A.

So, the correct answer is A. 1/4 K.