A horizontal spring with k=35N/m is compressed .085m and used to launch a .075kg marble. find the marble's launch speed. Repeat for a vertical launch.
Use conservation of energy.
Compressed spring potential energy = Launch kinetic energy
(1/2) k X^2 = (1/2) M Vo^2
X is the amount of spring compression.
Solve for Vo.
Vo = X*sqrt(k/M)
The launch speed will be nearly the same for a vertical or horizontal launch, but you will need to account for gravitation potential energy change, M g X, when the spring releases the marble in the vertical case.
To find the launch speed of the marble using a horizontal spring, we can use the principle of conservation of mechanical energy.
1. Horizontal Launch:
When the spring is compressed, it stores potential energy, which will be converted into kinetic energy of the marble when released. The formula for potential energy stored in a spring is given by:
Potential Energy (PE) = (1/2) * k * x^2
where k is the spring constant and x is the compression of the spring.
Given:
k = 35 N/m (spring constant)
x = 0.085 m (compression of the spring)
Calculating the potential energy:
PE = (1/2) * 35 N/m * (0.085 m)^2
Now, calculate the kinetic energy (KE) of the marble. The initial potential energy stored in the spring is fully converted into kinetic energy when the marble is released.
KE = PE
Substituting the values:
KE = (1/2) * 35 N/m * (0.085 m)^2
To calculate the launch speed, we need to know the mass (m) of the marble. Given that m = 0.075 kg,
KE = (1/2) * m * v^2
Where v is the launch speed we need to find.
Now, we can solve for v by equating the kinetic energy and solving for v:
KE = (1/2) * m * v^2
(1/2) * 35 N/m * (0.085 m)^2 = (1/2) * 0.075 kg * v^2
Cancelling out the common factors and solving for v, we have:
v^2 = (35 N/m * (0.085 m)^2) / 0.075 kg
v = sqrt((35 N/m * (0.085 m)^2) / 0.075 kg)
Evaluating the expression to find v gives:
v ≈ 0.845 m/s
Therefore, the marble's launch speed using a horizontal spring is approximately 0.845 m/s.
2. Vertical Launch:
When the spring is launched vertically, we need to consider the potential energy converted into gravitational potential energy as the marble reaches its highest point.
Using the same calculations as the horizontal launch, we find the potential energy stored in the spring:
PE = (1/2) * 35 N/m * (0.085 m)^2
To find the launch speed, we also need to consider the gravitational potential energy as the marble reaches its highest point. At this point, the potential energy stored in the spring is fully converted into gravitational potential energy:
PE = m * g * h
where g is the acceleration due to gravity (9.8 m/s^2) and h is the maximum height reached by the marble.
By equating the potential energy from the spring to the gravitational potential energy:
PE = m * g * h
(1/2) * 35 N/m * (0.085 m)^2 = 0.075 kg * 9.8 m/s^2 * h
Solving for h:
h = (1/2) * 35 N/m * (0.085 m)^2 / (0.075 kg * 9.8 m/s^2)
Substituting the given values and calculating h, we have:
h ≈ 0.034 m
Now that we know the maximum height reached by the marble, we can find the launch speed using the formula for final velocity in free fall:
v = sqrt(2 * g * h)
Substituting the values and evaluating:
v = sqrt(2 * 9.8 m/s^2 * 0.034 m)
v ≈ 0.785 m/s
Therefore, the marble's launch speed using a vertical spring is approximately 0.785 m/s.
To find the launch speed of the marble in both horizontal and vertical launches, we can use the principle of conservation of mechanical energy.
1. Horizontal Launch:
In a horizontal launch, the spring potential energy is converted into kinetic energy. The formula to calculate the launch speed is:
KE = (1/2)mv^2
where KE is the kinetic energy, m is the mass of the marble, and v is the launch speed.
To find the kinetic energy, we need to calculate the potential energy stored in the spring. The formula for potential energy of a spring is:
PE = (1/2)kx^2
where PE is the potential energy, k is the spring constant, and x is the compression or extension of the spring.
Given:
k = 35 N/m (spring constant)
x = 0.085 m (compression)
m = 0.075 kg (mass of the marble)
First, let's calculate the potential energy of the compressed spring:
PE = (1/2)kx^2
PE = (1/2)(35 N/m)(0.085 m)^2
PE = 0.127 N
Now, we can equate the potential energy to the kinetic energy:
PE = KE
0.127 N = (1/2)mv^2
Rearranging the equation, we can solve for v:
v^2 = (2 * 0.127 N) / m
v^2 = (2 * 0.127 N) / 0.075 kg
v^2 = 3.39 N/kg
v ≈ √3.39 ≈ 1.84 m/s
Therefore, the marble's launch speed in the horizontal direction is approximately 1.84 m/s.
2. Vertical Launch:
In a vertical launch, both the spring potential energy and the gravitational potential energy are converted into kinetic energy. The formula to calculate the launch speed is the same as before:
KE = (1/2)mv^2
To find the kinetic energy, we need to calculate the total potential energy stored in the spring and the gravitational potential energy. The formula for gravitational potential energy is:
PEgravity = mgh
where PEgravity is the gravitational potential energy, m is the mass of the marble, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height of the launch.
Given:
m = 0.075 kg (mass of the marble)
g = 9.8 m/s² (acceleration due to gravity)
h = x = 0.085 m (vertical displacement)
First, let's calculate the potential energy of the compressed spring:
PEspring = (1/2)kx^2
PEspring = (1/2)(35 N/m)(0.085 m)^2
PEspring = 0.127 N
Now, let's calculate the gravitational potential energy:
PEgravity = mgh
PEgravity = (0.075 kg)(9.8 m/s²)(0.085 m)
PEgravity = 0.064 N
Total potential energy:
PE = PEspring + PEgravity
PE = 0.127 N + 0.064 N
PE = 0.191 N
Equating potential energy to kinetic energy:
PE = KE
0.191 N = (1/2)mv^2
Rearranging the equation, we can solve for v:
v^2 = (2 * 0.191 N) / m
v^2 = (2 * 0.191 N) / 0.075 kg
v^2 = 5.09 N/kg
v ≈ √5.09 ≈ 2.26 m/s
Therefore, the marble's launch speed in the vertical direction is approximately 2.26 m/s.