Question

A 195-lb man standing on a surface of negligible friction kicks forward a \(0.158\) -lb stone lying at his feet so that it acquires a speed of \(12.7 \mathrm{ft} / \mathrm{s}\). What velocity does the man acquire as a result?

Short answer

The velocity acquired by the man as a result of the kick is approximately -0.0103 ft/s, in the direction opposite to the stone's motion.

Step-by-step solution

Set Up the Momentum Conservation Equation

According to the principle of conservation of momentum, the total momentum of a system before and after an event (in this case the kick) must remain the same. Therefore, the initial momentum (before the kick) must be equal to the final momentum (after the kick). Considering only the horizontal direction (as all movements are horizontal), the equation of conservation of momentum can be written as, \(m_1 \cdot v_1 + m_2 \cdot v_2 = m_1 \cdot v_1' + m_2 \cdot v_2'\) where \(m_1\) and \(m_2\) are the masses of the man and the stone respectively, \(v_1\) and \(v_2\) are their initial velocities (which are zero before the kick), and \(v_1'\) and \(v_2'\) are their final velocities.

Substitute Known Values

Now we substitute the known values into the equation. According to the question, the man's mass \(m_1\) is 195 lbs, the stone's mass \(m_2\) is 0.158 lbs, the stone's final velocity \(v_2'\) is 12.7 ft/s, and \(v_1\) and \(v_2\) are zero (since both the man and the stone were initially at rest). The man's final velocity \(v_1'\) is the unknown we want to find. So we can put these values into the equation. After substitution, the equation will be 0 + 0 = 195 lbs \cdot v_1' + 0.158 lbs \cdot 12.7 ft/s.

Solve for the Unknown

From step 2 we need to solve the equation to get the final velocity of the man \(v_1'\). Simplifying the equation, we get \(v_1'\) = - (0.158 lbs \cdot 12.7 ft/s) / 195 lbs. The negative sign indicates that the direction of man's movement is opposite to that of the stone.

Key concepts

Physics Problem Solving

When tackling any physics problem, especially one involving conservation laws, a structured approach is key. Begin by understanding the scenario completely. In problems like the one about a man kicking a stone, identify the physical quantities involved—masses, velocities, and directions of motion.

Applying the principle of conservation of momentum requires analyzing the system in its entirety. Start by identifying what's interacting within the system. Here, it's the man and the stone. Visualize how momentum is transferred between them. This helps in setting up your equations accurately.

Next, use equations that represent the conservation laws. Clearly arrange your known and unknown variables before solving them. Breaking complex problems into systematic steps can simplify decision-making and calculation processes.

Momentum Equation

Momentum, represented as the product of mass and velocity, plays a crucial role in understanding motion. The equation for momentum is given by: \( p = m \cdot v \), where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity. When two objects interact, their total momentum before and after the interaction remains constant, provided no external forces act on them. This is the core idea behind the conservation of momentum.

In our scenario, initially, both the man and the stone are at rest. Thus, their total initial momentum is zero. After the man kicks the stone, both acquire different velocities. The conservation of momentum equation becomes:

• Initial Momentum = Final Momentum
• \( m_1 \cdot 0 + m_2 \cdot 0 = m_1 \cdot v_1' + m_2 \cdot v_2' \)

Substitute known values to find unknown velocities. Remember, this equation helps balance and solve for unknowns by equating the momentum on both sides.

Velocity Analysis

Velocity is key to determining the movement direction and speed of any object. In problems like these, it's essential to assess how different actions affect velocity. Here, the kick transfers some of the man's momentum to the stone. The stone gains a velocity of \(12.7\; \text{ft/s} \), and in return, the man gains a velocity backward.

To find the man's final velocity, rearrange the momentum equation:

• From \(195 \cdot v_1' = -0.158 \cdot 12.7\), solve \(v_1'\).
• This results in \(v_1' = - \frac{(0.158 \cdot 12.7)}{195} \), which calculates the exact speed and direction.

The negative velocity indicates the opposite direction to the stone's movement, emphasizing action-reaction pairs.