Question

A spherical ball of mass \(15 \mathrm{~kg}\) stationary at the top of a hill of height \(82 \mathrm{~m} .\) It slides down a smooth surface to the ground, then climbs up another hill of height \(32 \mathrm{~m}\) and finally slides down to horizontal base at a height of \(10 \mathrm{~m}\) above the ground. The velocity attained by the ball is (A) \(30 \sqrt{10 \mathrm{~m} / \mathrm{s}}\) (B) \(10 \sqrt{30 \mathrm{~m} / \mathrm{s}}\) (C) \(12 \sqrt{10} \mathrm{~m} / \mathrm{s}\) (D) \(10 \sqrt{12} \mathrm{~m} / \mathrm{s}\)

Short answer

The velocity attained by the ball is approximately \(10\sqrt{30}\) m/s.

Step-by-step solution

Total Energy at the Initial Height

At first, we need to analyze the total energy of the ball at the initial height (82 meters). At this height, the gravitational potential energy is maximum, and the kinetic energy is zero as the ball is stationary. The potential energy PE can be calculated as follows: PE = m × g × h Here, m = 15 kg (mass of the ball) g = 9.81 m/s² (acceleration due to gravity) h = 82 m (initial height) PE = 15 kg × 9.81 m/s² × 82 m PE = 12015 J (Joules)

Energy Conservation between First Hill and Second Hill

As the ball rolls down the first hill and climbs the second hill, we can use the conservation of energy law to calculate the velocity of the ball after passing the second hill height (32 meters). We can assume the Potential Energy at the height of 10 meters is PE10 = m × g × 10 Moreover, the Potential Energy at the height of 32 meters is PE32 = m × g × 32 The Kinetic energy KE10 and KE32 at heights 10 meters and 32 meters can be calculated using the following equation: KE10 = PE - PE10 KE32 = PE - PE32 Since the velocity is the same at 10 meters and 32 meters height, the Kinetic energies must be equal: KE10 = KE32 Therefore, the Potential Energies at these heights should follow: PE10 - PE = PE32 - PE

Calculate the Potential Energy at Heights

Using the equations from Step 2, we can calculate the Potential Energy at different heights: PE10 = m × g × 10 PE10 = 15 kg × 9.81 m/s² × 10 m PE10 = 1471.5 J PE32 = m × g × 32 PE32 = 15 kg × 9.81 m/s² × 32 m PE32 = 4716 J

Calculate the Kinetic Energy and the Final Velocity

After calculating the Potential Energies, we can calculate the Kinetic Energy and then the final velocity: KE10 = PE - PE10 KE10 = 12015 J - 1471.5 J KE10 = 10543.5 J Since KE10 = KE32, we have the Kinetic Energy at the height of 32 meters: KE32 = 10543.5 J Now, we calculate the velocity of the ball using the relation KE = (1/2)mv²: 10543.5 J = (1/2) × 15 kg × v² Dividing both sides by 7.5 kg, we get: v² = 1405.8 Taking the square root of both sides: v = \(\sqrt{1405.8}\) m/s v ≈ 37.5 m/s Comparing the values to the given options, the final velocity is closest to \(10\sqrt{30}\) m/s, which is option (B).

Key concepts

Gravitational Potential Energy

Understanding gravitational potential energy (GPE) is key to solving problems like the one described. When an object is at a height, it has stored energy due to its position in the gravitational field. This energy is known as gravitational potential energy. To calculate it, use the formula:

• \( \text{PE} = m \times g \times h \)

Here, \(m\) is the mass of the object, \(g\) is the acceleration due to gravity (\(9.81 \, \text{m/s}^2\)), and \(h\) is the height above the ground.

In our exercise, the ball starts at a height of 82 meters. With a mass of 15 kg, we can determine the gravitational potential energy at this point, which totals \(12015 \, \text{J}\). This indicates the maximum potential energy available before it begins to move, converting this energy into kinetic or other forms as it travels down the hill.

Kinetic Energy Calculation

Kinetic energy (KE) shows how much energy an object possesses due to its motion. As the ball in our exercise moves down the hill, its gravitational potential energy converts into kinetic energy. The formula for kinetic energy is:

• \( \text{KE} = \frac{1}{2} m v^2 \)

Where \(v\) is the velocity of the object.

When solving for the ball's velocity, we need to know its kinetic energy first. By looking at the values determined in the exercise, at a lower height when potential energy decreases, kinetic energy increases. This energy conversion gives us the tool to find the velocity of the ball as it travels, specifically at the base height of 10 meters.
We learned that at this height, kinetic energy is calculated as \(10543.5 \, \text{J}\). With this, apply the kinetic energy formula to solve for \(v\), leading us to the ball’s velocity result of approximately \(37.5 \, \text{m/s}\). This conversion demonstrates energy conservation and the transition between potential and kinetic energy.

Physics Problem Solving Steps

Solving physics problems involves systematic breaking down of information and applying established laws. Let's talk about the steps typically followed in energy-related problems.

• Start by identifying known quantities, such as mass, heights, and initial conditions.
• Use the conservation of energy principle. Energy cannot be created or destroyed, only transformed. Here, it means the total energy stays constant as the ball moves.
• Calculate the gravitational potential energy at various points using \( \text{PE} = m \times g \times h \).
• Determine changes in potential energy as the ball moves from one height to another.
• Find the kinetic energy from the potential energy difference at various points.
• Use \( \text{KE} = \frac{1}{2} m v^2 \) to calculate velocities, ensuring energy conservation is maintained.

These steps allow you to analyze the motion and energy conversions of objects sliding down hills, making it simpler to predict outcomes such as speed at various positions. Applying these steps leads to efficient, correct solutions in physics challenges.