The drawing shows two boxes resting on frictionless ramps. One box is relatively light and sits on a steep ramp. The other box is heavier and rests on a ramp that is less steep. The boxes are released from rest at A and allowed to slide down the ramps. The two boxes have masses of 12 and 38 kg. If A and B are 5.5 and 1.5 m, respectively, above the ground, determine the speed of (a) the lighter box and (b) the heavier box when each reaches B. (c) What is the ratio of the kinetic energy of the heavier box to that of the lighter box at B?

(a)

ΔKE=ΔPE
KE(B)-KE( A)=PE(A)-PE(B)
mv²/2 -0 =mgh(A)-mgh(B)
v=sqrt{2g[h(A)-h(B)]}

(b) KE1/KE2 = {m1•gh(A)-mgh(B)}/ {m2•gh(A)-mgh(B)} =
=m1/m2

To solve this problem, we can use the principles of conservation of energy and the laws of physics. Let's break down the problem into smaller steps:

Step 1: Determine the potential energy at point A for both boxes.
The potential energy can be calculated using the formula: potential energy = mass * acceleration due to gravity * height. Since we have the masses and heights of both boxes, we can calculate their potential energies at point A.

Potential energy of the lighter box:
PE₁ = m₁ * g * h₁
where m₁ = mass of the lighter box, g = acceleration due to gravity, h₁ = height of point A.

Potential energy of the heavier box:
PE₂ = m₂ * g * h₂
where m₂ = mass of the heavier box, h₂ = height of point A.

Step 2: Calculate the potential energy at point B for both boxes.
Since both boxes are sliding down the ramps without friction, their potential energy will be converted into kinetic energy at point B. This means that the potential energy at point B for both boxes will be zero.

PE₁ = KE₁ + 0
PE₂ = KE₂ + 0

Step 3: Calculate the kinetic energy at point B for both boxes.
The kinetic energy can be calculated using the formula: kinetic energy = (1/2) * mass * velocity². Since the potential energy is converted into kinetic energy, we can equate the potential energy at point A to the kinetic energy at point B.

KE₁ = PE₁
KE₂ = PE₂

Step 4: Calculate the velocities of both boxes at point B.
Using the equations derived in Step 3, we can determine the velocities of the boxes at point B.

velocity of the lighter box:
KE₁ = (1/2) * m₁ * v₁²
v₁ = √((2 * KE₁) / m₁)

velocity of the heavier box:
KE₂ = (1/2) * m₂ * v₂²
v₂ = √((2 * KE₂) / m₂)

Step 5: Calculate the ratio of the kinetic energy of the heavier box to that of the lighter box at point B.
The ratio of the kinetic energies can be calculated by dividing the kinetic energy of the heavier box by the kinetic energy of the lighter box.

Ratio = KE₂ / KE₁

By following these steps and plugging in the given values for masses and heights, you can calculate the speeds of the two boxes and the ratio of their kinetic energies at point B.