Question

5.60 A man throws a rock of mass \(m=0.325 \mathrm{~kg}\) straight up into the air. In this process, his arm does a total amount of work \(W_{\text {net }}=115 \mathrm{~J}\) on the rock. Calculate the maximum distance, \(h\), above the man's throwing hand that the rock will travel.

Short answer

Answer: The maximum height the rock will reach above the man's hand is approximately 3.6 meters.

Step-by-step solution

Calculate the initial kinetic energy of the rock

To find the maximum height, we first need the initial kinetic energy of the rock. We know the net work done on the rock, \(W_{\text{net}}\), and can use the work-energy principle to find the initial kinetic energy, \(K_1\). The principle states that the work done on an object is equal to the change in kinetic energy, therefore, \(W_{\text {net}}=K_1 - K_0\). Since the rock is initially at rest, \(K_0 = 0\). So, the initial kinetic energy is equal to the net work done: \(K_1 = W_{\text {net}} = 115\ \mathrm{J}\).

Calculate the maximum potential energy of the rock

At the maximum height, the rock will momentarily come to rest before falling back down, which means its final kinetic energy, \(K_2\), will be zero. Therefore, all of its initial kinetic energy will convert into potential energy, \(U_2\), due to gravity. Using the work-energy principle again, \(K_1 - K_2 = U_2 - U_1\). Since the rock is initially at the man's hand height, we can consider its initial potential energy, \(U_1\), to be zero. Thus, the maximum potential energy is equal to the initial kinetic energy: \(U_2 = K_1 = 115\ \mathrm{J}\).

Calculate the maximum height of the rock above the man's hand

Now, we can use the formula for gravitational potential energy, \(U=mgh\), to find the maximum height, \(h\). At the maximum height, the potential energy will be equal to the initial kinetic energy, so \(U_2 = mgh\). Rearranging this equation, we get: \(h = \frac{U_2}{mg}\) Substitute the given values for \(m\), \(U_2\), and \(g=9.8\ \mathrm{m/s^2}\) (the acceleration due to gravity): \(h = \frac{115\ \mathrm{J}}{0.325\ \mathrm{kg} * 9.8\ \mathrm{m/s^2}}\)

Calculate the maximum height

Finally, perform the calculation: \(h \approx 3.6\ \mathrm{m}\) Therefore, the maximum distance above the man's throwing hand that the rock will travel is approximately 3.6 meters.

Key concepts

Potential Energy

Potential energy is the energy stored in an object due to its position relative to other objects. It's like the energy a rock has when sitting at the top of a hill. This energy has the potential to do work, which could be moving the rock down the hill. When the potential energy is released, it can cause movement or change.
In everyday life, potential energy can be found in multiple ways:

• A stretched spring stores potential energy, which can make it snap back to its original shape when released.
• A drawn bow stores potential energy before it launches an arrow.
• Food holds chemical potential energy, ready to be used by our bodies for movement and activity.

In physics, potential energy is often associated with forces such as gravity or elasticity. This form of energy plays a crucial role in analyzing how energy is converted from one type to another, such as turning into kinetic energy when an object starts moving.

Kinetic Energy

Kinetic energy is the energy an object possesses because of its motion. Anytime you see an object moving, it has kinetic energy. The faster something moves, the more kinetic energy it has. This energy type depends not only on speed but also on mass. A car traveling at a certain speed has more kinetic energy than a smaller, lighter bike going at the same speed.
The formula to calculate kinetic energy is:
\[ K = \frac{1}{2} mv^2 \]
Where:

• \( K \) is the kinetic energy,
• \( m \) is the mass of the object,
• \( v \) is the velocity of the object.

As seen in the problem, when the rock is thrown upwards, the arm of the man does work on it, giving it kinetic energy. This energy allows the rock to oppose gravity and ascend until it reaches its highest point, where it's briefly at rest with kinetic energy converting entirely to potential energy before it begins its descent.

Gravitational Potential Energy

Gravitational potential energy is a type of potential energy related to an object’s position in a gravitational field. When you raise an object higher up from its original position, it gains gravitational potential energy. The greater the height, the more potential energy it has. This is due to the gravitational force acting on the mass of the object, which tends to pull it back down.
The formula for gravitational potential energy is:
\[ U = mgh \]
Where:

• \( U \) is the gravitational potential energy,
• \( m \) is the mass of the object,
• \( g \) is the acceleration due to gravity, usually \( 9.8 \text{ m/s}^2 \),
• \( h \) is the height above the reference point.

As the rock in the exercise reaches its maximum height, the kinetic energy initially given to it by the throw converts fully into gravitational potential energy. This energy determines how long and how high the rock will stay elevated in the air before gravity pulls it back down. In the given problem, this change in forms of energy helps us figure out the maximum height the rock reaches.