With a mass = 10 k
.. the horizontal surface on which the block slides is frictionless. The speed of the block before it touches the spring is 6.0 m/s. How fast is he block moving at the instant the spring has been compressed 15 cm? K=2.0kN/m
To find the speed of the block at the instant the spring has been compressed 15 cm, we can use the principle of conservation of mechanical energy. The initial kinetic energy of the block will be equal to the final potential energy stored in the compressed spring.
Here's the step-by-step process to find the speed of the block:
1. Convert the mass of the block to kilograms:
Mass = 10 kg
2. Convert the compression of the spring from centimeters to meters:
Compression = 15 cm = 0.15 m
3. Find the potential energy stored in the spring:
Potential Energy = (1/2) * k * x^2
Where k is the spring constant and x is the compression of the spring.
Spring constant (k) = 2.0 kN/m
Convert kN to N: 2.0 kN = 2.0 * 1000 N
Substitute the values into the formula:
Potential Energy = (1/2) * (2.0 * 1000 N/m) * (0.15 m)^2
4. Now we need to find the initial kinetic energy of the block when it touches the spring. We can use the equation:
Initial Kinetic Energy = (1/2) * m * v^2
Where m is the mass of the block and v is its initial velocity.
Mass = 10 kg
Initial velocity (v) = 6.0 m/s
5. Substitute the values into the equation to find the initial kinetic energy.
6. Since mechanical energy is conserved, we can equate the initial kinetic energy to the final potential energy:
Initial Kinetic Energy = Potential Energy
Solve this equation to find the final velocity of the block.
By following these steps and performing the calculations, you should obtain the value for the speed of the block at the instant the spring has been compressed 15 cm.
To find the speed of the block when the spring has been compressed, we can use the principle of conservation of mechanical energy. The initial mechanical energy of the block, which includes its kinetic energy, will be equal to its final mechanical energy when the spring is compressed.
1. Determine the initial kinetic energy (KE) of the block:
The kinetic energy (KE) of an object is given by the formula: KE = (1/2) * mass * velocity^2
Given:
Mass (m) = 10 kg
Initial velocity (v) = 6.0 m/s
Substituting the given values into the formula, we have:
KE_initial = (1/2) * 10 kg * (6.0 m/s)^2 = 180 J
2. Determine the potential energy (PE) of the spring when it is compressed:
The potential energy stored in a spring is given by the formula: PE = (1/2) * k * x^2
Given:
Spring constant (k) = 2.0 kN/m = 2000 N/m (since 1 kN = 1000 N)
Compression distance (x) = 15 cm = 0.15 m
Substituting the given values into the formula, we have:
PE_spring = (1/2) * 2000 N/m * (0.15 m)^2 = 22.5 J
3. Determine the final kinetic energy (KE) of the block when the spring is compressed:
Since mechanical energy is conserved, the final kinetic energy of the block will be equal to the initial kinetic energy minus the potential energy stored in the spring:
KE_final = KE_initial - PE_spring = 180 J - 22.5 J = 157.5 J
4. Find the final velocity (v) of the block:
To find the final velocity, we need to convert the final kinetic energy into velocity. Using the same formula as step 1, we can solve for the final velocity:
KE_final = (1/2) * mass * velocity^2
Substituting the known values into the formula, we have:
157.5 J = (1/2) * 10 kg * velocity^2
15.75 = velocity^2
velocity = √15.75 ≈ 3.97 m/s
Therefore, the block is moving at approximately 3.97 m/s at the instant the spring has been compressed 15 cm.