Question

A steel ball falls from a stationary air balloon. The mass of the ball is 0,3kg. It strikes the ground with a vertical download velocity of 60m/s. Calculate
a) The height through which the steel ball falls
b) the potential energy of the steel ball when it starts falling.
c) the kinetic energy of the steel ball when in 50m above ground level
d) The total mechanical energy the steel ball has 20m above ground level

Answer 1

To solve these questions, we'll use the law of conservation of mechanical energy, which states that the sum of potential energy and kinetic energy remains constant in the absence of non-conservative forces like friction.

a) The height through which the steel ball falls can be calculated using the equation:
Potential energy at the starting point (when it's at rest) = Kinetic energy when it hits the ground.
Pe = Ke
mgh = 0.5mv^2

Here, m = mass of the ball = 0.3 kg
g = acceleration due to gravity = 9.8 m/s^2 (approx.)
v = velocity of the ball when it hits the ground = 60 m/s

Substituting these values, we get:
0.3 * 9.8 * h = 0.5 * 0.3 * 60^2
2.94h = 540
h = 183.67 m

Therefore, the height through which the steel ball falls is approximately 183.67 meters.

b) The potential energy of the steel ball when it starts falling can be calculated using the equation:
Potential energy = mass * gravity * height

Here, m = mass of the ball = 0.3 kg
g = acceleration due to gravity = 9.8 m/s^2 (approx.)
h = height through which the ball falls = 183.67 m (calculated in part a)

Potential energy = 0.3 * 9.8 * 183.67
Potential energy = 538.93 J

Therefore, the potential energy of the steel ball when it starts falling is approximately 538.93 Joules.

c) The kinetic energy of the steel ball when it is 50 m above ground level can be calculated using the same equation:
Kinetic energy = 0.5 * mass * velocity^2

Here, m = mass of the ball = 0.3 kg
v = vertical download velocity of the ball = 60 m/s

Kinetic energy = 0.5 * 0.3 * 60^2
Kinetic energy = 540 J

Therefore, the kinetic energy of the steel ball when it is 50 m above ground level is 540 Joules.

d) The total mechanical energy the steel ball has 20 m above ground level can be calculated by summing the potential and kinetic energy at that height.

Potential energy = mass * gravity * height
Kinetic energy = 0.5 * mass * velocity^2

Here, m = mass of the ball = 0.3 kg
g = acceleration due to gravity = 9.8 m/s^2 (approx.)
h = height = 20 m
v = vertical download velocity = ?

Using the same equation for potential energy as in part b, we can solve for v.

0.3 * 9.8 * 20 = 0.5 * 0.3 * v^2
5.88 = 0.045 * v^2
v^2 = 5.88 / 0.045
v^2 = 130.67
v = √130.67
v = approximately 11.43 m/s

Now we can calculate the total mechanical energy:

Potential energy = 0.3 * 9.8 * 20 = 58.8 J
Kinetic energy = 0.5 * 0.3 * (11.43)^2 = 19.49 J

Total mechanical energy = Potential energy + Kinetic energy
Total mechanical energy = 58.8 + 19.49
Total mechanical energy = approximately 78.29 J

Therefore, the total mechanical energy the steel ball has 20 m above ground level is approximately 78.29 Joules.