Question

a 0.04-kg ball is thrown from the top of a 30-m tall building (point a) at an unknown angle above the horizontal. as shown in the figure, the ball attains a maximum height of 10 m above the top of the building before striking the ground at point b. if air resistance is negligible, what is the value of the kinetic energy of the ball at b minus the kinetic energy of the ball at a (kb - ka)?

Answer 1

The kinetic energy at point B (Kb) minus the kinetic energy at point A (Ka) is determined by first calculating the potential energy at point A and assuming it converts to kinetic energy at point B due to the conservation of energy, resulting in Kb - Ka = 15.68 J - unknown initial KE at A.

Step-by-step explanation:

To determine the value of the kinetic energy of the ball at point B minus the kinetic energy of the ball at point A (Kb - Ka), we need to apply the principles of energy conservation.

Calculate the potential energy at point A:

The potential energy (PE) at point A can be calculated using the formula PE = mgh, where m is mass, g is the acceleration due to gravity (9.8 m/s²), and h is the height. Since the ball is at a height of 30 m plus the additional 10 m (the maximum height above the building), the total height (h) is 40 m.

PE at A = (0.04 kg) × (9.8 m/s²) × (40 m) = 15.68 J

Calculate the kinetic energy at point B:

At point B, just before striking the ground, all the potential energy at the highest point (point A) would have been converted to kinetic energy (KE) due to energy conservation in the absence of air resistance. The kinetic energy at point B can be calculated using the formula KE = 0.5 × m × v², where v is the velocity of the ball at point B, which can be found using the principle that the potential energy at point A equals the kinetic energy at point B (PE at A = KE at B).

15.68 J = 0.5 × (0.04 kg) × v²

v² = (15.68 J) / (0.02 kg) = 784 m²/s²

v = √784 m²/s² = 28 m/s

KE at B = 0.5 × (0.04 kg) × (28 m/s)² = 15.68 J

Calculate Kb - Ka:

Since at point A the ball is thrown, it already has some kinetic energy in addition to its potential energy, but without the initial velocity we can't calculate the exact kinetic energy at A. However, for the difference Kb - Ka, we can say that the kinetic energy at B is purely from the conversion of its potential energy at A.

So, Kb - Ka = KE at B - KE at A

Answer 2

The problem requires calculating the change in kinetic energy of a ball thrown from a building using conservation of energy. The potential energy at the beginning and the kinetic energy when the ball hits the ground both need to be calculated, and the increase in kinetic energy can then be determined.

Step-by-step explanation:

The student's question involves calculating the change in kinetic energy of a ball thrown from the top of a building, at the topmost point of its trajectory (point A), and when it strikes the ground (point B).

To solve this problem, we use the principles of conservation of mechanical energy and apply the formulas for potential energy and kinetic energy.

a. The potential energy of the ball at point A (just as it's thrown) can be calculated using the formula PE = mgh, where m is mass, g is the acceleration due to gravity, and h is the height. Here, PE = (0.04 kg) * (9.8 m/s2) * (30 m).

b. To calculate the kinetic energy of the ball at point B, we need to consider that the ball has fallen an additional 10 meters beyond the height of the building.

Hence, the total distance fallen is h = 40 meters, and the potential energy at the top would have been converted into kinetic energy at point B, which can be calculated as KE = 0.5 * m * v2.

To find v, we can use the conservation of energy or the kinematic equation v2 = u2 + 2gh, where u is the initial velocity (which is 0 at the top of the building for the vertical component).

c. The maximum velocity of the ball is reached just before it strikes the ground and is equal to the square root of 2gh, factoring in the height of the building and the additional height it reached beyond the building.

In this case, the height is 40 meters because the ball rose 10 meters above the building and then fell the entire 40 meters.