/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 55 The cart has a mass of \(3 \math... [FREE SOLUTION] | 91影视

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Chapter 15: Problem 55

The cart has a mass of \(3 \mathrm{~kg}\) and rolls freely from \(A\) down the slope. When it reaches the bottom, a spring loaded gun fires a \(0.5-\mathrm{kg}\) ball out the back with a horizontal velocity of \(v_{b / c}=0.6 \mathrm{~m} / \mathrm{s}\), measured relative to the cart. Determine the final velocity of the cart.

Short Answer

Expert verified
The final velocity of the cart is \(0.1 \mathrm{m/s}\), in the opposite direction to the initial motion of the cart.

Step by step solution

01

State the Momentum Conservation Principle

The momentum conservation principle states that the total momentum of an isolated system remains constant if no external forces act on it. For this exercise, the system is the cart and the ball. Since no external horizontal forces are acting on this system, the horizontal momentum before the ball is fired is equal to the horizontal momentum after the ball is fired. Mathematically, this is stated as \(m_{c}v_{c} = m_{c}v_{c}'+ m_{b}v_{b/c}'\), where \(m_c\) is the mass of the cart, \(v_c\) is the initial velocity of the cart, \(v_c'\) is the final velocity of the cart, \(m_b\) is the mass of the ball, and \(v_{b/c}'\) is the velocity of the ball relative to the cart after being fired.
02

Plug in Known Values

We are given that \(m_c = 3 \, \mathrm{kg}\), \(m_b = 0.5 \, \mathrm{kg}\), and \(v_{b/c}' = -0.6 \, \mathrm{m/s}\). The negative sign represents that the ball is moving in the opposite direction to the initial motion of the cart. Also, the cart rolls freely from A down the slope, therefore its initial velocity \(v_c = 0\). Substituting these values, we get \[0 = 3v_c' + -0.5 \cdot (0.6)\].
03

Solve for Final Velocity of Cart

Finally, solving for \(v_c'\) gives us the final velocity of the cart. Reorganizing the equation results in \(v_c' = \frac{-0.5 \cdot (0.6)}{3}\).

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