Question

A ball is allowed to fall from a height \(20 \mathrm{~m}\). If there is \(30 \%\) loss of energy due to impact, then after one impact ball will go up to (A) \(18 \mathrm{~m}\) (B) \(16 \mathrm{~m}\) (C) \(12 \mathrm{~m}\) (D) \(14 \mathrm{~m}\)

Short answer

After one impact with a 30% loss of energy, the ball will bounce back up to a height of \(14 \mathrm{~m}\). The answer is (D) \(14 \mathrm{~m}\).

Step-by-step solution

Remember the formula for gravitational potential energy

To solve this problem, we need to recall the formula for gravitational potential energy: \[PE = mgh\] where: - PE is the potential energy, - m is the mass of the ball, - g is the acceleration due to gravity (approximately \(9.81 \mathrm{~m/s^2}\)), - h is the height from the surface.

Calculate the initial potential energy of the ball

When the ball is at a height of 20 meters, it has gravitational potential energy, which we can calculate using the formula from step 1. We can ignore the mass as it will cancel out later: \[PE_1 = mgh = mg(20)\]

Calculate the potential energy after impact

Since there is a 30% loss of energy during the impact, we need to calculate 70% of the initial potential energy to find the remaining potential energy after impact: \[PE_2 = 0.7 * PE_1 = 0.7 * mg(20)\]

Calculate the new height after the impact

Now we need to find the new height the ball will reach after the impact. To do this, we set the new potential energy (PE_2) equal to the gravitational potential energy formula: \[PE_2 = mgh_2\] Since we know the value of PE_2, we can substitute it in and solve for the new height (h_2): \[0.7 * mg(20) = mgh_2\] Divide both sides by mg to get: \[0.7 * 20 = h_2\]

Calculate the new height of the ball

Now solve for the new height h_2: \[h_2 = 0.7 * 20 = 14\] So, after one impact with a 30% loss of energy, the ball will bounce back up to a height of 14 meters. The answer is (D) \(14 \mathrm{~m}\).

Key concepts

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy stored in an object due to its position above the ground. This form of energy depends on three key factors:

• The mass of the object (m)
• The height (h) from a zero level, typically the earth’s surface
• The gravitational acceleration (g), which on Earth is approximately 9.81 m/s²

The formula for calculating gravitational potential energy is given by \[ PE = mgh \]Knowing this helps us understand why a ball held at a height has potential energy stored within it. If dropped, this energy can convert to kinetic energy as the ball falls. The potential energy decreases as the ball drops because the height off the ground decreases. Gravitational potential energy plays a key part in predicting how high a ball will bounce back after hitting the ground.

Impact and Energy Conservation

Energy conservation in physics states that energy cannot be created or destroyed. Instead, it transforms from one type to another. For our bouncing ball, as it drops, the gravitational potential energy is mainly converted into kinetic energy. However, during the impact, not all energy is conserved as mechanical energy. Some is lost, often due to heat and sound.
In this exercise, the problem mentions a 30% loss in energy during the impact. This means that 70% of the original energy remains as potential energy after the bounce.
To compute the potential energy left after impact, we use:\[ PE_{after} = 0.7 \times PE_{before} \]This energy loss affects how high the ball can bounce back. Even though energy is not completely conserved as mechanical energy, the idea helps us estimate and calculate the resulting behavior after the collision. Analyzing energy changes gives us insights into how forces work during impacts and help us assess energy efficiency.

Height Calculation after Impact

After an impact, not all the gravitational potential energy is converted back into motion. Instead, the ball rises to a new, lower height. We already calculated that 70% of the original energy is conserved. This directly corresponds to how high the ball will reach on its bounce.
To determine the new height, we relate the remaining energy post-impact back to the potential energy formula. We've outlined it using,\[ PE_{after} = mgh_{new} \]By substituting,\[ 0.7 \times mgh_{original} = mgh_{new} \]dividing by mass and gravitational acceleration gives us the new height,\[ h_{new} = 0.7 \times h_{original} \]In our current example, it becomes:\[ h_{new} = 0.7 \times 20 \]\[ h_{new} = 14 \text{ meters} \]Thus, the ball will reach a height of 14 meters after the bounce, showcasing a practical application of energy conservation in real-world problems.