A 4kg and a 2kg block are moving along a horizontal frictionless surface. The 4 kg block is moving to the right with a speed of 6 m/s and the 2 kg block is moving to the left with a speed of 3 m/s. The two blocks collide inelastically and then continue to move together along the surface and then up a 30° frictionless ramp. How high up the ramp does the combined object move
Use conservation of momentum to get the final velocity of the stuck-together blocks. Then compute the new total kinetic energy using that velocity.
For the final step, use conservation of energy. Initial KE of the stuck blocks equals the potential energy gain at their highest point they get to on the ramp.
To find the height that the combined object moves up the ramp, we need to analyze the conservation of momentum and conservation of energy.
First, let's determine the final velocity of the combined object after the collision. Since the collision is inelastic, the two blocks stick together and move as one.
Using the principle of conservation of momentum:
Initial momentum of the 4 kg block = mass × velocity = 4 kg × 6 m/s = 24 kg⋅m/s to the right
Initial momentum of the 2 kg block = mass × velocity = 2 kg × (-3 m/s) = -6 kg⋅m/s to the left
Total initial momentum = 24 kg⋅m/s - 6 kg⋅m/s = 18 kg⋅m/s to the right
Since the blocks stick together, their total mass is 4 kg + 2 kg = 6 kg.
Therefore, the final velocity of the combined object is:
Final velocity = Total initial momentum / Total mass = 18 kg⋅m/s / 6 kg = 3 m/s to the right
Next, let's determine the height the combined object moves up the ramp using the principle of conservation of energy.
The initial kinetic energy of the combined object is:
Initial kinetic energy = 1/2 × mass × (velocity)^2
= 1/2 × 6 kg × (3 m/s)^2
= 27 J
Since the surface is frictionless, there is no work done against friction. Thus, all the initial kinetic energy is converted to potential energy.
The potential energy at height h on the ramp is given by:
Potential energy = mass × gravity × height × cos(angle)
= 6 kg × 9.8 m/s^2 × h × cos(30°)
Setting the potential energy equal to the initial kinetic energy, we have:
27 J = 6 kg × 9.8 m/s^2 × h × cos(30°)
Simplifying the equation:
h = 27 J / (6 kg × 9.8 m/s^2 × cos(30°))
Calculating the height:
h ≈ 1.02 m
Therefore, the combined object moves approximately 1.02 meters up the ramp.