A 2 kg mass falls 40 m and buries itself into the ground 0.40 m. Find: a) How much work was done by gravity, b) How much potential energy was lost, and c) What was its speed when it hit the ground, and d) the average force causing the mass to stop once it hit the ground.
In order to answer these questions, we need to consider the principles of work, potential energy, and kinetic energy.
a) To calculate the work done by gravity, we can use the formula:
Work = Force × Distance × cos(θ)
Since gravity acts vertically downward, the angle (θ) between the force and distance is 0 degrees, and the formula simplifies to:
Work = Force × Distance
The force of gravity is given by the weight of the object, which is:
Weight = mass × acceleration due to gravity
Acceleration due to gravity is approximately 9.8 m/s².
Therefore, the work done by gravity is:
Work = Weight × Distance = (mass × acceleration due to gravity) × Distance
b) Potential energy is associated with an object's position relative to a reference point. The potential energy lost by the object can be calculated using the formula:
Potential Energy = Weight × Height
In this case, the height is the distance over which the object falls and buries itself into the ground.
c) The speed of the object when it hits the ground can be calculated using the principle of conservation of mechanical energy. The initial potential energy of the object is converted into kinetic energy as it falls. At the point of impact, all potential energy is converted into kinetic energy. The kinetic energy can be calculated using the formula:
Kinetic Energy = 0.5 × mass × velocity²
Equating the initial potential energy to the final kinetic energy:
Potential Energy = Kinetic Energy
Solving for velocity, we get:
velocity = sqrt((2 × Potential Energy) / mass)
d) The average force causing the mass to stop once it hits the ground can be calculated using Newton's second law of motion. The average force is equal to the change in momentum divided by the time interval taken to stop the object. Assuming the stopping time is very short and the mass is not rebounding, we can calculate average force as:
Average Force = Change in momentum / Time interval
Change in momentum = Final momentum - Initial momentum
Since the mass comes to a complete stop, the final momentum is zero. The initial momentum can be calculated using the formula:
Initial momentum = mass × initial velocity
Therefore, the average force is:
Average Force = (mass × initial velocity) / Time interval
To obtain the specific numerical values, we need the initial velocity and the time interval taken to stop the object.