In the figure, a 2.4 kg box of running shoes slides on a horizontal frictionless table and collides with a 1.4 kg box of ballet slippers initially at rest on the edge of the table, at height h = 0.75 m. The speed of the 2.4 kg box is 4.6 m/s just before the collision. If the two boxes stick together because of packing tape on their sides, what is their kinetic energy just before they strike the floor?

To find the kinetic energy just before the two boxes strike the floor, we need to calculate the total kinetic energy of both boxes.

The kinetic energy of an object can be calculated using the formula:

KE = (1/2) * m * v^2

where KE is the kinetic energy, m is the mass, and v is the velocity.

Given information:
Mass of the running shoes box (m1) = 2.4 kg
Mass of the ballet slippers box (m2) = 1.4 kg
Initial velocity of running shoes box (v1) = 4.6 m/s

1. Calculate the kinetic energy of the running shoes box just before the collision:
KE1 = (1/2) * m1 * v1^2
= (0.5) * 2.4 kg * (4.6 m/s)^2

2. Calculate the kinetic energy of the ballet slippers box initially at rest:
KE2 = 0 (since the ballet slippers box is at rest)

3. Calculate the total kinetic energy just before the collision:
Total KE = KE1 + KE2

4. Since the two boxes stick together after the collision, the total kinetic energy just before striking the floor is equal to the total kinetic energy just before the collision.

Therefore, the kinetic energy just before the two boxes strike the floor is equal to the total kinetic energy calculated in step 3.

To find the kinetic energy just before the boxes strike the floor, we need to calculate the total kinetic energy of both boxes before the collision.

The kinetic energy (KE) is given by the formula:

KE = (1/2) * mass * velocity^2

For the 2.4 kg box of running shoes, the mass is 2.4 kg and the velocity is 4.6 m/s. Plugging these values into the formula:

KE1 = (1/2) * 2.4 kg * (4.6 m/s)^2

Similarly, for the 1.4 kg box of ballet slippers initially at rest, the velocity is 0 (since it is at rest), so the kinetic energy is:

KE2 = (1/2) * 1.4 kg * (0 m/s)^2 = 0

Now, when the two boxes stick together, their velocities become the same after the collision. Let's call this velocity v (which is the final velocity). Since they stick together, we can treat them as a single combined object with a total mass of 2.4 kg + 1.4 kg = 3.8 kg.

The kinetic energy just before they strike the floor is the sum of the initial kinetic energy of the running shoes box and the ballet slippers box:

KE_total = KE1 + KE2

Substituting the values:

KE_total = (1/2) * 2.4 kg * (4.6 m/s)^2 + 0

Now, we can calculate the value of KE_total.

m₁=2.4 kg, m₂=1.4 kg, v₁=4.6 m/s, v₂=0

m₁v₁= (m₁+m₂) u
u= m₁v₁/(m₁+m₂) =2.4•4.6/(2.4+1.4) =2.9 m/s
At the edge of the table, two boxes have
kinetic (KE) and potential energy (PE), and near the ground - KE₁
KE + PE=KE₁
(m₁+m₂)u²/2 + (m₁+m₂)gh =(m₁+m₂)v²/2
u²/2 +gh = v²/2
v=sqrt{ u²+2gh} =sqrt{2.9² +2•9.8•0.75}=3.65 m/s