A cart of mass

m1 = 12 kg
slides down a frictionless ramp and is made to collide with a second cart of mass
m2 = 21 kg,
which then heads into a vertical loop of radius 0.26 m, but the carts stick together after the collision.Find the height h from which cart 1 would need to start to make sure that both carts complete the loop without leaving the track.

To find the height from which cart 1 would need to start to ensure that both carts complete the loop without leaving the track, we can use the principle of conservation of energy.

First, we need to determine the speed at the top of the loop by considering the conservation of mechanical energy. At the top of the loop, the total mechanical energy of the system is the sum of the kinetic energy (KE) and the gravitational potential energy (PE) of the two carts.

The initial kinetic energy is given by:
KEi = (1/2) m1 v1^2

At the top of the loop, the kinetic energy is converted into gravitational potential energy:
PEf = (m1 + m2) g h

The final gravitational potential energy is given by:
PEf = (m1 + m2) g (2r)

Since the carts stick together after the collision, their combined mass becomes (m1 + m2).

Setting the initial kinetic energy equal to the final gravitational potential energy, we have:
(1/2) m1 v1^2 = (m1 + m2) g (2r)

Rearranging the equation, we can solve for the initial velocity v1:
v1^2 = 2g (m1 + m2) r
v1 = √(2g (m1 + m2) r)

Now, to find the height h from which cart 1 should start, we can consider the conservation of mechanical energy from the starting height to the top of the loop.

The initial mechanical energy is given by:
Ei = (1/2) m1 v1^2 + m1gh

The final mechanical energy at the top of the loop is:
Ef = (m1 + m2) gh

Setting the initial mechanical energy equal to the final mechanical energy, we have:
(1/2) m1 v1^2 + m1gh = (m1 + m2) gh

Substituting the value of v1 we found earlier, we can solve for h:
(1/2) m1 (2g (m1 + m2) r) + m1gh = (m1 + m2) gh

Simplifying the equation, we get:
m1 (m1 + m2) r + 2m1h = (m1 + m2) h

Rearranging the equation, we can solve for h:
h = m1 (m1 + m2) r / (2m1 + m2)

Plugging in the given values:
m1 = 12 kg
m2 = 21 kg
r = 0.26 m

We can now calculate the height h:
h = 12 kg * (12 kg + 21 kg) * 0.26 m / (2 * 12 kg + 21 kg)

Calculating this expression will give us the required height h from which cart 1 should start to ensure that both carts complete the loop without leaving the track.