A student uses a spring to launch a marble vertically in the air. The mass of the marble is 0.002 kg and when the spring is stretched 0.05 m it exerts a force of 10 N. What is the maximum height the marble can reach?
A student uses a spring loaded launcher to launch a marble vertically in the air. The mass of the marble is 0.003 kg and the spring constant is 249 N/m. What is the maximum height the marble can reach when compressed 7 cm?
To find the maximum height the marble can reach, we can use the law of conservation of mechanical energy. At the maximum height, all the initial kinetic energy of the marble will be converted into potential energy.
Step 1: Calculate the potential energy at the maximum height
The potential energy (PE) at the maximum height can be given by the formula:
PE = m * g * h
where m is the mass of the marble (0.002 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the maximum height.
PE = 0.002 kg * 9.8 m/s² * h
Step 2: Calculate the initial kinetic energy
The initial kinetic energy (KE) can be given by the formula:
KE = 1/2 * m * v²
where v is the velocity of the marble at the point of release.
Step 3: Equate the potential energy and initial kinetic energy
Since the law of conservation of mechanical energy states that energy is conserved, we can equate the potential energy and initial kinetic energy:
PE = KE
0.002 kg * 9.8 m/s² * h = 1/2 * 0.002 kg * v²
Simplifying the equation:
4.9 h = v²
Step 4: Calculate the velocity of the marble at the point of release
To calculate the velocity, we need to find the amount of potential energy stored in the spring at a displacement of 0.05 m.
Potential energy (PE_spring) stored in a spring is given by the formula:
PE_spring = 1/2 * k * x²
where k is the spring constant and x is the displacement of the spring.
Given that the force exerted by the spring is 10 N when it is stretched by 0.05 m, we can calculate the spring constant:
10 N = k * 0.05 m
k = 10 N / 0.05 m
k = 200 N/m
The potential energy stored in the spring at a displacement of 0.05 m can be calculated as:
PE_spring = 1/2 * 200 N/m * (0.05 m)²
PE_spring = 0.5 J
Since energy is conserved, this potential energy will be converted into kinetic energy.
0.5 J = 1/2 * 0.002 kg * v²
Step 5: Solve for the velocity
0.5 J = 0.001 kg * v²
v² = 0.5 J / 0.001 kg
v² = 500 m²/s²
v = √(500 m²/s²) ≈ 22.36 m/s
Step 6: Calculate the maximum height
Using the equation from Step 3:
4.9 h = v²
4.9 h = (22.36 m/s)²
4.9 h = 499.3936 m²/s²
h = 499.3936 m²/s² / 4.9
h ≈ 101.8 m
Therefore, the maximum height the marble can reach is approximately 101.8 meters.
To find the maximum height the marble can reach, we need to use the concept of mechanical energy.
The mechanical energy of the system remains constant throughout the motion of the marble. It consists of two components: potential energy (due to the height) and kinetic energy (due to the velocity).
At the maximum height, the marble's kinetic energy is zero because it comes to a stop before falling back down. Therefore, all the initial energy from the spring is converted into potential energy.
The potential energy (PE) is given by the equation:
PE = m * g * h
where m is the mass of the marble, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.
Now let's calculate the potential energy:
PE = 0.002 kg * 9.8 m/s^2 * h
To find the height (h), we need to first find the spring constant (k) for the spring:
k = F / x
where F is the force exerted by the spring (10 N in this case) and x is the displacement of the spring (0.05 m in this case).
k = 10 N / 0.05 m
k = 200 N/m
The equation for the potential energy of a spring is given by:
PE = (1/2) * k * x^2
Since all the initial potential energy from the spring is converted into potential energy at the maximum height, we can equate the two equations:
(1/2) * k * x^2 = m * g * h
Substituting the values:
(1/2) * 200 N/m * (0.05 m)^2 = 0.002 kg * 9.8 m/s^2 * h
Simplifying:
0.005 J = 0.0196 kg.m/s^2 * h
Now, solve for h:
h = 0.005 J / (0.0196 kg.m/s^2)
h ≈ 0.255 m
Therefore, the maximum height the marble can reach is approximately 0.255 meters.