Question

A \(110-\mathrm{kg}\) man lowers himself to the ground from a height of \(12 \mathrm{~m}\) by holding on to a rope passed over a frictionless pulley and attached to a 74 -kg sandbag. ( \(a\) ) With what speed does the man hit the ground? (b) Is there anything he could do to reduce the speed with which he hits the ground?

Short answer

Solution: (a) The speed of the man when he hits the ground can be calculated in step 2 of the solution. (b) To reduce his descent speed, the man could apply a downward force on the rope or let go of the rope entirely.

Step-by-step solution

Determine the total mechanical energy before the man starts to move

At the moment before the man starts to lower himself down, he possesses potential energy due to his height above the ground and the sandbag too has potential energy due to its position lifted up. As the system is at rest, kinetic energy is zero. So, the initial total mechanical energy of the system can be calculated as: \( E_{i} = m_{1}gh_{1} + m_{2}gh_{2} \) where \( m_{1} \) and \( m_{2} \) are the mass of the man and sandbag respectively, \( g \) is the acceleration due to gravity, and \( h_{1} \) and \( h_{2} \) are the respective heights above the ground. Since the sandbag is initially resting on the ground, \( h_{2} = 0 \) . Thus, \( E_{i} = m_{1}gh_{1} \). Plugging in the given values: \( E_{i} = (110 ~\mathrm{kg})(9.8 ~\mathrm{m/s^2})(12 ~\mathrm{m}) \).

Determine the total mechanical energy as the man hits the ground

As the man hits the ground and the sandbag reaches its maximum height, the system's total mechanical energy will be the sum of the kinetic energy of the man and the potential energy of the sandbag. Hence, \( E_{f} = (1/2) m_{1}v_{1}^{2} + m_{2}gh_{2}^{f} \), where \( v_{1} \) is the speed of the man and \( h_{2}^{f} \) is the final height of the sandbag. Making use of the mechanical energy conservation principle (\( E_{i} = E_{f} \)), we can solve for \( v_{1} \) to find the speed of the man as he hits the ground.

Reducing the speed of impact

The reduction in the speed with which the man hits the ground can be achieved by increasing the kinetic energy of the sandbag as the man hits the ground. This could be done by the man letting go of the rope, thus transferring his potential energy into the kinetic energy of the sandbag. It is also noteworthy that by applying a downward force on the rope while descending, he would be doing work on the system which can also reduce his descent speed.

Key concepts

Potential Energy

Imagine standing on a ladder holding a basketball. As you hold the ball away from the ground, it has energy purely by virtue of its position relative to the Earth's surface. This energy is what we call potential energy, specifically gravitational potential energy. It's present anytime an object is positioned above the ground. The higher the object, or the greater its mass, the more potential energy it has. In our exercise, the man and the sandbag both possess potential energy when they are above the ground. The formula for gravitational potential energy is given by \( U = mgh \), where \( m \) stands for mass, \( g \) is the acceleration due to gravity (\(9.8 \text{ m/s}^2\) on Earth), and \( h \) is the height above the ground. For the man in the exercise, his potential energy before descending is significant because of his considerable mass and height.

Kinetic Energy

Now, imagine dropping that basketball from the ladder. As it falls, the ball moves faster and faster, gaining speed due to gravity. The energy associated with this movement is kinetic energy. For any moving object, kinetic energy is given by the expression \( KE = \frac{1}{2}mv^2 \), where \( m \) is mass and \( v \) is velocity. In our scenario, as the man descends, his potential energy decreases while his kinetic energy increases because he's speeding up towards the earth. By the time he reaches the ground, all of the potential energy due to his height is converted into kinetic energy reflective of his speed, assuming no energy is lost to forces such as air resistance.

Frictionless Pulley

The textbook problem introduces a frictionless pulley as part of the system. A pulley is a simple machine that changes the direction of force applied to it. In reality, pulleys can have some amount of friction which can do work on the system, potentially changing the total mechanical energy. However, in our idealized case, a 'frictionless pulley' means that it does not add or remove any energy from the system. It serves just to redirect the forces without any energy loss. This is a crucial assumption because it underpins the conservation of mechanical energy within the system, making it possible to track the energy transfer from potential to kinetic without worrying about energy losses due to the pulley.

Work-Energy Principle

The work-energy principle relates the work done on an object to its change in kinetic energy. Work is a form of energy transfer and is done when a force causes an object to move. The work done by an external force is equal to the change in kinetic energy of the object on which the force is acting. For example, when the man in the exercise begins to descend, he does work against gravity. If he applies an additional force, such as pulling down on the rope as he goes down, he does extra work on the system, effectively reducing his own kinetic energy upon impact with the ground. This is directly tied to the concept that he could reduce the impact speed by doing work on the sandbag, increasing its kinetic energy at the point of impact.