A 0.150 kg block moving vertically upward collides with a light vertical spring and compresses it 4.50cm before coming to rest.
If the spring constant is 54.0 N/m, what was the initial speed of the block? (Ignore energy losses to sound and other factors during the collision.)
Thanks for including the dimensions (kg, etc) of the physical quantities this time.
When the spring is fully compressed, the initial kinetic energy (1/2)MV^2 is converted to gravitational energy MgY and spring potential energy (1/2)kY
Y is the vertical distance the spring is compressed (0.045 m)
g is the acceleration of gravity (9.81 m/s^2)
M is the 0.150 kg mass
k is the 54.0 N/m spring constant
(1/2)MV^2 = MgY + (1/2)kY^2
V^2 = 2gY + (k/M)Y^2
Solve for V by plugging in the numbers. The 2gY gravity term may turn out to be negligible compared to (k/M)Y^2
I just want to thank DRWLS. I sent to Cramster and the answer was wrong several times by them. Again thanks for help.
1.611m/s
To find the initial speed of the block, we can use the principle of conservation of mechanical energy.
The principle of conservation of mechanical energy states that the total mechanical energy of a system remains constant if no external forces are acting on it. In this case, since no external forces are mentioned, we can assume that the mechanical energy is conserved.
The mechanical energy of an object can be given by the sum of its kinetic energy and potential energy.
1. First, let's find the potential energy of the compressed spring.
The potential energy stored in a spring is given by the formula:
Potential energy (PE) = (1/2) * k * x^2
where k is the spring constant and x is the displacement of the spring from its equilibrium position.
Plugging in the given values:
k = 54.0 N/m
x = 4.50 cm = 0.045 m (convert cm to m)
PE = (1/2) * 54.0 N/m * (0.045 m)^2
= 0.054 J
2. Next, let's find the initial kinetic energy of the block.
The kinetic energy of an object can be given by the formula:
Kinetic energy (KE) = (1/2) * m * v^2
where m is the mass of the object and v is its velocity.
We are given the mass of the block: m = 0.150 kg
Let's assume the initial velocity of the block (upward) is v.
The final velocity of the block will be 0 as it comes to rest after compressing the spring.
KE = (1/2) * 0.150 kg * v^2
3. According to the conservation of mechanical energy, the initial kinetic energy is equal to the potential energy of the compressed spring.
Therefore, we can set the two expressions equal to each other and solve for v:
KE = PE
(1/2) * 0.150 kg * v^2 = 0.054 J
Simplifying the equation:
0.075 kg * v^2 = 0.054 J
v^2 = (0.054 J) / (0.075 kg)
v^2 = 0.72 m^2/s^2
Taking the square root of both sides, we get:
v = √(0.72 m^2/s^2)
v ≈ 0.85 m/s
Therefore, the initial speed of the block was approximately 0.85 m/s.