A 5 kg mass is thrown up with a velocity of 4 m/s from a height of 30 m onto a spring with a relaxed length of 10 m, and a constant of 400 N/m. What will the maximum compression of the spring be? What will the speed of the object be when the spring is compressed 0.3 m?
Have you tried applying conservation of energy? Show your work and one of us will gladly critique it.
(V2^2)=(Vi)^2+2ad
0=16-19.6d
-16=-19.6d
d=0.816 m + 30 m = 30.816 meters above the ground
mgh = mgh + 1/2kx^2 = (5 kg)(9.8 m/s^2)(30.816) = 200x^2-49x+490
x=2.75 m
mgh=1/2kx^2+mgh+1/2mv^2
(5)(9.8)(30.186)=1/2(400)(0.3)^2+(5)(9.8)(9.7)+1/2(5v^2)
v=20.17 m/s
To find the maximum compression of the spring, we need to first calculate the potential energy of the mass when it reaches its maximum height, and then equate it to the potential energy stored in the compressed spring.
1. Calculate the potential energy at maximum height:
Potential Energy (PE) = mass × gravitational acceleration × height
PE = 5 kg × 9.8 m/s² × 30 m = 1470 J
2. Calculate the potential energy stored in the compressed spring:
Potential Energy (PE) = 0.5 × spring constant × compression²
PE = 0.5 × 400 N/m × compression²
Since the potential energy at maximum height is equal to the potential energy stored in the spring, we can equate the two equations:
1470 J = 0.5 × 400 N/m × compression²
Now, solve for the compression:
compression² = (2 × 1470 J) / (400 N/m)
compression² = 7.35 m²
compression ≈ √7.35 ≈ 2.71 m
Therefore, the maximum compression of the spring will be approximately 2.71 meters.
To find the speed of the object when the spring is compressed 0.3 m, we can use the principle of conservation of mechanical energy. At this point, the potential energy of the mass is partially converted into the kinetic energy of the mass.
3. Calculate the potential energy stored in the spring when it is compressed 0.3 m:
Potential Energy (PE) = 0.5 × spring constant × compression²
PE = 0.5 × 400 N/m × (0.3 m)² = 18 J
4. Calculate the remaining kinetic energy:
Remaining Kinetic Energy (K.E.) = Initial Potential Energy - Potential Energy stored in the spring
Remaining K.E. = 1470 J - 18 J = 1452 J
5. Calculate the speed of the object using the remaining kinetic energy:
Remaining K.E. = 0.5 × mass × velocity²
1452 J = 0.5 × 5 kg × velocity²
Solve for the velocity:
velocity² = (2 × 1452 J) / (5 kg)
velocity² = 582.4 m²/s
velocity ≈ √582.4 ≈ 24.13 m/s
Therefore, the speed of the object when the spring is compressed 0.3 m will be approximately 24.13 m/s.