Two blocks are initially held at rest on 30° frictionless ramps . The 4 kg block is 1.25 m from the base of the ramp and the 3 kg block is 5 m above the base of the ramp. The two blocks are then released and slide down the ramps and onto a frictionless horizontal surface. The two blocks collide elastically with each other and then move backward up the ramps again. The 4 kg block comes to a stop 3.09 m up its ramp. How far up the ramp does the 3 kg move before coming to a stop?
Since every process is elastic or frictionless, the total potential energy must be the same when the colliding and sliding are over, even though the heights may change for each block. Use that fact to compute the final height of the 3kg block, above the horizontal base. That and the ramp angle will tell you how far it slid.
To find the distance the 3 kg block moves up the ramp before coming to a stop, we first need to calculate the initial velocities of both blocks as they slide down the ramps.
We can begin by finding the gravitational potential energy of each block at their respective starting positions:
Potential energy (PE) = mass (m) × acceleration due to gravity (g) × height (h)
For the 4 kg block:
PE = 4 kg × 9.8 m/s^2 × 1.25 m = 49 J
For the 3 kg block:
PE = 3 kg × 9.8 m/s^2 × 5 m = 147 J
Since both blocks start at rest, all of their potential energy will be converted into kinetic energy (KE) as they slide down the ramps. We can calculate the kinetic energy using the formula:
KE = (1/2) × mass (m) × velocity squared (v^2)
For the 4 kg block:
49 J = (1/2) × 4 kg × v^2
v^2 = 49 J / (2 × 4 kg)
v^2 = 6.125 m^2/s^2
v = sqrt(6.125 m^2/s^2) ≈ 2.47 m/s
For the 3 kg block:
147 J = (1/2) × 3 kg × v^2
v^2 = 147 J / (2 × 3 kg)
v^2 = 24.5 m^2/s^2
v = sqrt(24.5 m^2/s^2) ≈ 4.95 m/s
Now that we have the initial velocities, we can calculate the distance the 3 kg block moves up the ramp before coming to a stop.
Using the conservation of mechanical energy for the 3 kg block:
Potential energy (PE) + Kinetic energy (KE) = 0
At the stopping point, all of the block's mechanical energy is converted back into potential energy:
PE = m × g × h
Potential energy at the stopping point = 3 kg × 9.8 m/s^2 × height
We want to find the height (h) the 3 kg block reaches when it stops. So we rewrite the equation above as:
h = (Potential energy at the stopping point) / (m × g)
= (PE / (3 kg × 9.8 m/s^2)
Plugging in the known values:
h = (147 J) / (3 kg × 9.8 m/s^2)
h ≈ 5 m
Therefore, the 3 kg block moves approximately 5 meters up the ramp before coming to a stop.