A body of mass 1/4 kg falls from rest through a height of 40 m and comes to rest having penetrated a distance of 1/2 m into the sandy ground . Calculate the average force exerted by the sand in bringing the body to rest
To solve this problem, we can use the work-energy principle. The work-energy principle states that the work done on an object is equal to its change in kinetic energy.
The work done by the gravitational force as the body falls through a height of 40 m is given by:
Work_gravity = mgh
= (1/4 kg)(9.8 m/s^2)(40 m)
= 98 J
The work done by the sand in bringing the body to rest is equal to the work done by the average force exerted by the sand multiplied by the distance the body penetrates into the ground. We can calculate this work using the formula:
Work_sand = Force_sand x distance_penetrated
To find the average force exerted by the sand, we need to know the final velocity of the body just before it comes to rest. We can find this using the kinematic equation:
vf^2 = vi^2 + 2ad
Where:
vf = final velocity (0 m/s, since the body comes to rest)
vi = initial velocity (unknown)
a = acceleration (acceleration due to gravity = 9.8 m/s^2)
d = distance traveled (1/2 m into the sandy ground)
Solving for vi, we get:
vi^2 = vf^2 - 2ad
= (0 m/s)^2 - 2(9.8 m/s^2)(-1/2 m)
= 4.9 m^2/s^2
Taking the square root of both sides, we find:
vi = 2.2 m/s
Now, using the work-energy principle:
Work_sand = Work_gravity
Force_sand x distance_penetrated = mgh
Force_sand = (mgh) / distance_penetrated
= (1/4 kg)(9.8 m/s^2)(40 m) / (1/2 m)
= 98 J / (1/2 m)
= 196 N
Therefore, the average force exerted by the sand in bringing the body to rest is 196 Newtons.
To calculate the average force exerted by the sand in bringing the body to rest, we need to use the concept of work done.
The work done on an object can be calculated using the formula:
Work = Force x Distance
In this case, the distance is the penetration of the body into the sandy ground, which is 1/2 m.
The work done on the body is equal to the change in its potential energy as it falls through the height of 40 m. The potential energy change is given by:
Potential Energy Change = m x g x h
where:
m = mass of the body = 1/4 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height = 40 m
Potential Energy Change = (1/4 kg) x (9.8 m/s^2) x (40 m)
= 98 J
Now, let's calculate the force exerted by the sand using the work-energy principle.
The work done by the sand force is equal to the negative of the potential energy change:
Work = -98 J
The average force exerted by the sand can be calculated by dividing the work by the distance:
Average Force = Work / Distance
Average Force = -98 J / 1/2 m
= -98 J / (1/2) m
= -196 J/m
Therefore, the average force exerted by the sand to bring the body to rest is -196 J/m. Note that the negative sign indicates that the force is acting in the opposite direction of displacement.