In compressing the spring in a toy dart gun, 0.5 J of work is done.
When the gun is fired, the spring gives its potential energy to a dart with a mass of
0.02 kg.
(a) What is the dart’s kinetic energy as it leaves the gun?
(b) What is the dart speed?
(a) The work W done compressing the spring becomes stored potential energy. When the spring is decompressed the same amount of energy be
becomes kinetic energy.
(b) Use the definition and value of kinetic energy to compute V
To find the dart's kinetic energy as it leaves the gun, we can use the formula:
Kinetic Energy = (1/2) * mass * velocity^2
Given:
Work done = 0.5 J
Mass of dart = 0.02 kg
(a) What is the dart's kinetic energy as it leaves the gun?
To find the kinetic energy, we need to find the velocity of the dart.
We can use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy.
So, the work done (0.5 J) is equal to the change in kinetic energy.
0.5 J = Final Kinetic Energy - Initial Kinetic Energy
Since the dart starts from rest (initially stationary), the initial kinetic energy is zero.
Therefore, the final kinetic energy is 0.5 J.
(b) What is the dart speed?
To find the dart's speed, we need to calculate the velocity using the kinetic energy and mass.
Kinetic Energy = (1/2) * mass * velocity^2
0.5 J = (1/2) * 0.02 kg * velocity^2
We can rearrange the equation to solve for velocity:
velocity^2 = (2 * 0.5 J) / 0.02 kg
velocity^2 = 50 J/kg
Taking the square root of both sides:
velocity = sqrt(50 J/kg)
Calculating the value:
velocity ≈ 7.07 m/s
Therefore, the dart's speed as it leaves the gun is approximately 7.07 m/s.
To find the answers to the given question, we can use the principle of conservation of energy. According to this principle, energy is neither created nor destroyed but is transferred from one form to another. In this case, the potential energy stored in the compressed spring is converted into the kinetic energy of the dart.
(a) The potential energy stored in the spring is given as 0.5 J. Since the potential energy is fully transferred to the dart, it becomes the kinetic energy of the dart as it leaves the gun.
Therefore, the dart's kinetic energy is also 0.5 J.
(b) The formula for kinetic energy is given by:
Kinetic Energy = (1/2) * mass * velocity^2
We can rearrange this formula to solve for velocity:
velocity = sqrt((2 * kinetic energy) / mass)
Substituting the values into the formula:
velocity = sqrt((2 * 0.5 J) / 0.02 kg)
velocity = sqrt(50 J / 0.02 kg)
velocity = sqrt(2500 m^2/s^2 / kg)
velocity ≈ 70.71 m/s
Therefore, the dart's speed as it leaves the gun is approximately 70.71 m/s.