Question

A crow drops a 0.11-kg clam onto a rocky beach from a height of \(9.8 \mathrm{~m}\). What is the kinetic energy of the clam when it is \(5.0 \mathrm{~m}\) above the ground? What is its speed at that point?

Short answer

The kinetic energy is 5.174 J and the speed is 9.67 m/s at 5.0 m above the ground.

Step-by-step solution

Calculate Initial Potential Energy

The initial potential energy (PE) when the clam is at 9.8 m is calculated using the formula: \(PE = mgh\), where \(m = 0.11\, \mathrm{kg}\), \(g = 9.8\, \mathrm{m/s^2}\), and \(h = 9.8\, \mathrm{m}\). Thus, \(PE = 0.11 \times 9.8 \times 9.8 = 10.564\, \mathrm{J}\).

Calculate Potential Energy at 5.0 m above Ground

To find the potential energy at 5.0 m, use the formula \(PE = mgh\) with \(h = 5.0\, \mathrm{m}\). This gives \(PE = 0.11 \times 9.8 \times 5.0 = 5.39\, \mathrm{J}\).

Determine Kinetic Energy at 5.0 m

The total mechanical energy is conserved, so the initial energy equals the energy at 5.0 m. Therefore, \(KE_{5m} = PE_{initial} - PE_{5m} = 10.564 - 5.39 = 5.174\, \mathrm{J}\).

Calculate Speed at 5.0 m using Kinetic Energy

The formula for kinetic energy is \(KE = \frac{1}{2}mv^2\). Solving for \(v\), we get \(v = \sqrt{\frac{2 \times KE}{m}}\). Substituting \(KE = 5.174\, \mathrm{J}\) and \(m = 0.11\, \mathrm{kg}\), \(v = \sqrt{\frac{2 \times 5.174}{0.11}} = 9.67\, \mathrm{m/s}\).

Key concepts

Potential Energy

Have you ever wondered why an object at a height holds energy without moving? This is due to potential energy, which is the energy stored due to an object's position relative to a gravitational field. To compute potential energy (\(PE\)), we use the formula:

• \( PE = mgh \)

Here,

• \(m\) is the mass of the object in kilograms,
• \(g\) is the acceleration due to gravity (approximately \(9.8 \, \mathrm{m/s^2}\) on Earth),
• \(h\) is the height above the ground in meters.

The higher an object is, the more potential energy it possesses.In the scenario with the clam, the initial potential energy when it is 9.8 m above ground is calculated using this formula, factoring in its mass and height. This way, we found that it stores approximately \(10.564 \, \mathrm{J}\) of energy.

Mechanical Energy Conservation

The principle of mechanical energy conservation is like tracking energy in its different forms.This principle states that in the absence of non-conservative forces, such as friction, the total mechanical energy of an object remains constant. This means that potential energy can be converted into kinetic energy and vice versa. The formula for mechanical energy conservation can be expressed as:

• \( PE_{initial} + KE_{initial} = PE_{final} + KE_{final} \)

In the problem with the clam, as it falls, potential energy decreases while kinetic energy increases, keeping the total mechanical energy the same at any point until it hits the ground.By knowing the initial potential energy and calculating potential energy at 5.0 m, we determined the kinetic energy at this point, showing how energy is transferred from one form to another.

Kinematics

Kinematics is the branch of physics dealing with motion without considering the forces that cause this motion. It helps us describe how objects move.In kinematics, understanding speed is crucial. When we want to know how fast something moves, we can use the kinetic energy to find it. For this, we rearrange the kinetic energy formula:

• \( KE = \frac{1}{2}mv^2 \)

Solving for speed \(v\), we get:

• \( v = \sqrt{\frac{2 \times KE}{m}} \)

For the clam dropped by the crow, once we calculated the kinetic energy at 5.0 m from the ground, we could find its speed at that height. This gives a velocity of approximately 9.67 m/s, showing how fast the clam is traveling through space at that moment.