A toy gun uses a spring to project a 5.3 g soft rubber sphere vertically. The spring constant is 8.0 N/m, the barrel of the gun is 15 cm long, and a constant frictional force of 0.032 N exists between barrel and projectile. How high does the projectile go if the spring was compressed 5.0 cm for this launch? For this, I need the problem solved step by step.
To find the height to which the projectile is launched, we need to consider the different forces acting on it and apply the principles of work, energy, and motion.
Step 1: Calculate the potential energy stored in the spring when it is compressed.
The potential energy stored in a spring is given by the formula:
PE = (1/2)kx^2
where PE is the potential energy, k is the spring constant, and x is the displacement of the spring.
Given:
k = 8.0 N/m
x = 5.0 cm = 0.05 m (converted to meters)
Substituting these values into the formula, we have:
PE = (1/2)(8.0 N/m)(0.05 m)^2
PE = 0.01 J (Joules)
Step 2: Calculate the work done by the spring in launching the projectile.
The work done by the spring is equal to the potential energy stored in it.
W = PE = 0.01 J
Step 3: Calculate the work done against friction.
The work done against friction can be calculated using the formula:
W = Fd
where W is the work done, F is the force, and d is the distance.
Given:
F = 0.032 N
d = 0.15 m (distance traveled along the barrel)
Substituting these values into the formula, we have:
W = (0.032 N)(0.15 m)
W = 0.0048 J
Step 4: Calculate the net work done on the projectile.
The net work done on the projectile is equal to the work done by the spring minus the work done against friction.
NetWork = W - W_friction
NetWork = 0.01 J - 0.0048 J
NetWork = 0.0052 J
Step 5: Calculate the gravitational potential energy at the maximum height.
At the maximum height, the kinetic energy of the projectile is zero, and all of the energy is in the form of gravitational potential energy.
Gravitational potential energy is given by:
PE_gravity = mgh
where PE_gravity is the gravitational potential energy, m is the mass of the projectile, g is the acceleration due to gravity, and h is the height.
Given:
m = 5.3 g = 0.0053 kg (converted to kilograms)
g = 9.8 m/s^2
Substituting these values into the formula, we have:
PE_gravity = (0.0053 kg)(9.8 m/s^2)(h)
Step 6: Equate the net work done on the projectile with the gravitational potential energy.
Since energy is conserved, the net work done on the projectile is equal to the gravitational potential energy at the maximum height.
NetWork = PE_gravity
0.0052 J = (0.0053 kg)(9.8 m/s^2)(h)
0.0052 J = (0.05194 kg·m/s^2)(h)
0.0052 J = (0.05194 N)(h)
h = 0.0052 J / 0.05194 N
h ≈ 0.1 m (rounded to one decimal place)
Therefore, the projectile reaches a height of approximately 0.1 meters.
To find the height the projectile reaches, we need to determine the initial energy stored in the spring and calculate the maximum height it can reach, considering the forces acting on it.
Step 1: Determine the potential energy stored in the spring.
The potential energy stored in a spring can be calculated using the formula:
Potential Energy (PE) = (1/2) * k * x^2
where:
PE is the potential energy stored in the spring
k is the spring constant
x is the displacement/compression of the spring
Given:
k = 8.0 N/m (spring constant)
x = 5.0 cm = 0.05 m (compression of the spring)
Plugging in these values:
PE = (1/2) * 8.0 N/m * (0.05 m)^2
PE = 0.01 J (joules)
Step 2: Determine the work done against friction.
The work done against friction is the product of the frictional force and the distance traveled. In this case, the distance traveled is the length of the barrel.
Given:
Frictional Force (f) = 0.032 N (frictional force)
Barrel Length (d) = 15 cm = 0.15 m (barrel length)
Work Done Against Friction (W) = f * d
W = 0.032 N * 0.15 m
W = 0.0048 J (joules)
Step 3: Calculate the initial kinetic energy of the projectile.
Since the potential energy is converted into kinetic energy, we can calculate the initial kinetic energy of the projectile using the principle of conservation of energy:
Initial Kinetic Energy (KE) = Potential Energy (PE) - Work Done Against Friction (W)
KE = 0.01 J - 0.0048 J
KE = 0.0052 J (joules)
Step 4: Determine the maximum height reached by the projectile.
At the maximum height, the projectile's kinetic energy is zero, and all its initial energy is converted into potential energy.
Potential Energy at Maximum Height (PE_max) = Initial Kinetic Energy (KE)
PE_max = 0.0052 J
Since potential energy is given by the formula:
PE = m * g * h
where:
m is the mass of the object (5.3 g = 0.0053 kg)
g is the acceleration due to gravity (approximately 9.8 m/s^2)
h is the maximum height reached by the object
Rearranging the formula:
h = PE_max / (m * g)
h = 0.0052 J / (0.0053 kg * 9.8 m/s^2)
h ≈ 0.976 m
Therefore, the projectile reaches a maximum height of approximately 0.976 meters.
The potential energy stored in the compressed spring, when the spring compression distance is X = 0.05 m, is
Ep = (1/2) k X^2 = (0.5)*8.0*(0.05)^2
= 0.0100 Joules
Make sure that the units of the spring constant are really N/m and not N/cm. That is a very small amount of stored spring energy.
The kinetic energy of the sphere leaving the gun equals the compressed spring potential energy MINUS the work done against friction. The friction work is 0.032 N * 0.15 m = 0.0048 J
That leaves
Ek = 0.0052 J for kinetic energy
0.0052 = (1/2) M V^2
Mass M = 0 0053 kg , so
V^2 = 2*.0052/0.0053 = 2
V = 1.4 m/s