QuestionJune 4, 2025

A car traveling at 29m/s runs out of gas while traveling up a 5.4^circ slope. (a) How far will it coast before starting to roll back down?

A car traveling at 29m/s runs out of gas while traveling up a 5.4^circ slope. (a) How far will it coast before starting to roll back down?
A car traveling at 29m/s runs out of gas while traveling up a 5.4^circ slope. (a) How far will it coast before starting to roll back down?

Solution
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Answer

456.07 m Explanation 1. Identify Forces and Energy The car will convert its kinetic energy into gravitational potential energy until it stops. Use **conservation of energy**: K.E. = P.E. 2. Write the Kinetic Energy Formula Initial kinetic energy: K.E. = \frac{1}{2}mv^2, where v = 29 \, \text{m/s}. 3. Write the Potential Energy Formula Potential energy at height h: P.E. = mgh, where g = 9.8 \, \text{m/s}^2. 4. Set Energies Equal and Solve for Height \frac{1}{2}mv^2 = mgh \implies h = \frac{v^2}{2g}. 5. Calculate Height h = \frac{(29)^2}{2 \times 9.8} = 42.91 \, \text{m}. 6. Relate Height to Distance Along Slope Use trigonometry: d = \frac{h}{\sin(\theta)}, where \theta = 5.4^\circ. 7. Calculate Distance d = \frac{42.91}{\sin(5.4^\circ)} = 456.07 \, \text{m}.

Explanation

1. Identify Forces and Energy<br /> The car will convert its kinetic energy into gravitational potential energy until it stops. Use **conservation of energy**: $K.E. = P.E.$<br /><br />2. Write the Kinetic Energy Formula<br /> Initial kinetic energy: $K.E. = \frac{1}{2}mv^2$, where $v = 29 \, \text{m/s}$.<br /><br />3. Write the Potential Energy Formula<br /> Potential energy at height $h$: $P.E. = mgh$, where $g = 9.8 \, \text{m/s}^2$.<br /><br />4. Set Energies Equal and Solve for Height<br /> $\frac{1}{2}mv^2 = mgh \implies h = \frac{v^2}{2g}$.<br /><br />5. Calculate Height<br /> $h = \frac{(29)^2}{2 \times 9.8} = 42.91 \, \text{m}$.<br /><br />6. Relate Height to Distance Along Slope<br /> Use trigonometry: $d = \frac{h}{\sin(\theta)}$, where $\theta = 5.4^\circ$.<br /><br />7. Calculate Distance<br /> $d = \frac{42.91}{\sin(5.4^\circ)} = 456.07 \, \text{m}$.
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