A 1000 kg cart is rolling to the right at 5.0 m/s. A 70 kg man is standing on the right end of the cart. What is the speed of the cart if the man suddenly starts running to the left with a speed of 10 m/s relative to the cart?

Verified step by step guidance

1. Identify the principle to use: This is a conservation of momentum problem. The total momentum of the system (cart + man) before and after the man starts running must remain the same because no external horizontal forces are acting on the system.
2. Define the variables: Let the mass of the cart be \( m_c = 1000 \; \text{kg} \), the mass of the man be \( m_m = 70 \; \text{kg} \), the initial velocity of the cart and man (together) be \( v_i = 5.0 \; \text{m/s} \), and the velocity of the man relative to the cart be \( v_{m/c} = -10 \; \text{m/s} \) (negative because the man runs to the left). Let the final velocity of the cart be \( v_f \).
3. Express the man's velocity relative to the ground: The velocity of the man relative to the ground is \( v_m = v_f + v_{m/c} \). Substituting \( v_{m/c} = -10 \; \text{m/s} \), we get \( v_m = v_f - 10 \; \text{m/s} \).
4. Write the conservation of momentum equation: The total momentum before the man starts running is \( (m_c + m_m) v_i \). The total momentum after the man starts running is \( m_c v_f + m_m v_m \). Set these equal: \( (m_c + m_m) v_i = m_c v_f + m_m v_m \).
5. Substitute known values and solve for \( v_f \): Replace \( v_m \) with \( v_f - 10 \; \text{m/s} \) in the equation. This gives \( (1000 + 70)(5.0) = 1000 v_f + 70(v_f - 10) \). Simplify and solve for \( v_f \) to find the final velocity of the cart.

Key Concepts

Conservation of Momentum

The principle of conservation of momentum states that in a closed system, the total momentum before an event must equal the total momentum after the event, provided no external forces act on it. In this scenario, the cart and the man form a closed system, and their combined momentum must remain constant as the man runs.

Relative Velocity

Relative velocity is the velocity of one object as observed from another object. In this problem, the man's speed is given relative to the cart, meaning we must account for both the cart's initial speed and the man's running speed to determine the final speed of the cart after the man starts running.

System Mass

The total mass of a system is crucial for calculating momentum. In this case, the system consists of the cart and the man. The combined mass affects how the momentum is distributed between the two when the man changes his velocity, impacting the final speed of the cart.