A mass of 2.0kg traveling at 3.0 m/s along a horizontal plane hits a relaxed spring 20cm away from it. The coefficient of kinetic friction is 0.3. The mass is slowed to zero velocity when the spring has been compressed. By how much the spring has been compressed, the spring constant of the spring is 800 N/m?
What I've done so far, is this right?
1/2mv^2 + mgh = 1/2mv^2 + mgh + 1/2kx^2
Ke - work done against friction = (1/2)kx^2
(1/2)mv^2 - .3 m g (.2+x) = (1/2)(800)x^2
(1/2)*2*9 -.3*2*9.8(.2+x) = 400 x^2
solve quadratic for x
why are you talking about m g h ??? it fell on horizontal plane?
Yes, you're on the right track with your equation, but there are a few things that need to be adjusted. Let's go through it step by step.
First, let's write down the given information:
- Mass of the object (m) = 2.0 kg
- Initial velocity (v) = 3.0 m/s
- Coefficient of kinetic friction (μ) = 0.3
- Distance from the object to the spring (s) = 20 cm = 0.20 m
- Spring constant (k) = 800 N/m
Next, let's establish the conservation of mechanical energy. The initial mechanical energy (KE + PE) of the object is equal to the final mechanical energy when it comes to rest after hitting the spring, plus the potential energy stored in the compressed spring.
The initial mechanical energy is given by:
Initial mechanical energy = 1/2 * m * v^2
The final mechanical energy is given by:
Final mechanical energy = m * g * h
In this case, h is the height of the object when it comes to rest after hitting the spring. We don't know the value of h yet.
The potential energy stored in the compressed spring is given by:
Potential energy of the spring = 1/2 * k * x^2
Here, x is the distance the spring is compressed, which is what we need to find.
Setting up the equation, we have:
1/2 * m * v^2 = (m * g * h) + 1/2 * k * x^2
Now, let's plug in the given values:
1/2 * 2.0 kg * (3.0 m/s)^2 = (2.0 kg * 9.8 m/s^2 * h) + 1/2 * 800 N/m * x^2
Simplifying this equation will help us solve for h and x.