A block with its mass M=200g is released from rest at a height of b=19 cm on a frictionless x=28° incline. It slides down the incline and then along a frictionless surface until it collides elastically with an block whose mass is 800 at rest d=1.5 m from the bottom of the incline. How much later do the two blocks collide again?
To calculate the time it takes for the two blocks to collide again, we'll need to break down the problem into a few steps:
Step 1: Calculate the acceleration of the first block sliding down the incline.
Since the incline is frictionless, we can use the formula for gravitational acceleration parallel to the incline:
a_parallel = g * sin(θ)
where g is the acceleration due to gravity (9.8 m/s²) and θ is the angle of the incline (28° in this case).
a_parallel = 9.8 m/s² * sin(28°)
Step 2: Calculate the time it takes for the first block to reach the bottom of the incline.
We can use the following kinematic equation to find the time:
s = ut + (1/2)at²
where s is the displacement, u is the initial velocity (0 since the block is released from rest), a is the acceleration, and t is the time.
s = b (height of the incline)
u = 0 m/s
a = a_parallel (from Step 1)
b = (1/2) * a * t²
Simplifying the equation:
2b = a * t²
t² = (2b) / a
t = sqrt((2b) / a)
Step 3: Calculate the time it takes for the first block to reach the second block.
We can use the equation for uniform motion to find the time:
s = ut + (1/2)at²
where s is the displacement (distance between the bottom of the incline and the second block), u is the initial velocity (calculated from Step 1), a is the acceleration (0 since the surface is frictionless), and t is the time.
s = d (distance between the bottom of the incline and the second block)
u = a_parallel (from Step 1)
a = 0 m/s²
d = u * t + (1/2) * a * t²
Since a = 0 m/s², the equation simplifies to:
d = u * t
Solving for t:
t = d / u
Step 4: Calculate the total time until the collision.
To find the total time, we sum the time it takes for the first block to reach the bottom of the incline (from Step 2) and the time it takes for the first block to reach the second block (from Step 3).
Total time = t (from Step 2) + t (from Step 3)
Now you can use the given values (M = 200 g, b = 19 cm, θ = 28°, d = 1.5 m) to calculate the total time until the two blocks collide again.