Ch 11: Impulse and Momentum

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 11: Impulse and Momentum

Problem 71b

Chapter 11, Problem 71b

A white ball traveling at 2.0m/s hits an equal-mass red ball at rest. The white ball is deflected by 25° and slowed to 1.5m/s. What percentage of the initial mechanical energy is lost in the collision?

Verified step by step guidance

Step 1: Start by calculating the initial kinetic energy of the system. The white ball is moving at 2.0 m/s, and the red ball is at rest. Use the formula for kinetic energy: $ KE = \frac{1}{2} m v^2 $. Since the red ball is at rest, its initial kinetic energy is zero. The total initial kinetic energy is therefore $ KE_{initial} = \frac{1}{2} m (2.0)^2 $.

Step 2: After the collision, the white ball is deflected by 25° and slowed to 1.5 m/s. Calculate the final kinetic energy of the white ball using the same formula: $ KE_{white, final} = \frac{1}{2} m (1.5)^2 $.

Step 3: The red ball, initially at rest, will now have some velocity after the collision. Let the velocity of the red ball after the collision be $ v_{red, final} $. Use the principle of conservation of momentum to find $ v_{red, final} $. Break the momentum into x and y components: $ m v_{white, initial} = m v_{white, final} \cos(25°) + m v_{red, final, x} $ for the x-direction, and $ 0 = m v_{white, final} \sin(25°) - m v_{red, final, y} $ for the y-direction.

Step 4: Solve the momentum equations to find the magnitude of $ v_{red, final} $, which is $ \sqrt{v_{red, final, x}^2 + v_{red, final, y}^2} $. Once you have $ v_{red, final} $, calculate the final kinetic energy of the red ball: $ KE_{red, final} = \frac{1}{2} m v_{red, final}^2 $.

Step 5: Add the final kinetic energies of both balls to find the total final kinetic energy: $ KE_{final} = KE_{white, final} + KE_{red, final} $. To find the percentage of mechanical energy lost, use the formula: $ \text{Percentage lost} = \frac{KE_{initial} - KE_{final}}{KE_{initial}} \times 100 $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conservation of Momentum

In a closed system, the total momentum before a collision is equal to the total momentum after the collision. This principle is crucial for analyzing collisions, as it allows us to set up equations based on the initial and final velocities of the objects involved. In this scenario, we can use the masses and velocities of the white and red balls to determine the final velocity of the red ball.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 0.5 * m * v², where m is mass and v is velocity. In this problem, we need to calculate the initial and final kinetic energies of both balls to determine the energy lost during the collision. Understanding how kinetic energy changes in elastic and inelastic collisions is essential for this analysis.

Energy Loss in Collisions

In inelastic collisions, some kinetic energy is transformed into other forms of energy, such as heat or sound, leading to a loss of mechanical energy. To find the percentage of energy lost, we compare the initial total kinetic energy to the final total kinetic energy after the collision. This concept is vital for understanding the efficiency of the collision and the extent of energy transformation that occurs.

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Related Practice

Textbook Question

In Problems $76 comma 77 comma 78 comma$ and $79$ you are given the equation(s) used to solve a problem. For each of these, you are to write a realistic problem for which this is the correct equation(s).

$\frac{1}{2} (0.30 kg) (0 m/s)^{2} + \frac{1}{2} (3.0 N/m) (Δ x_{2})^{2} = \frac{1}{2} (0.30 kg) (v_{1 x})^{2} + \frac{1}{2} (3.0 N/m) (0 m)^{2}$

Textbook Question

In Problems $76, 77, 78,$ and $79$ you are given the equation(s) used to solve a problem. For each of these, you are to finish the solution of the problem, including a pictorial representation.

$$\frac{1}{2}$ (0.30 $\text{ kg}$) (0 $\text{ m/s}$)^2 + $\frac{1}{2}$ (3.0 $\text{ N/m}$) ($\Delta$ x_2)^2 = $\frac{1}{2}$ (0.30 $\text{ kg}$) (v_{1x})^2 + $\frac{1}{2}$ (3.0 $\text{ N/m}$) (0 $\text{ m}$)^2$

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A 45 g projectile explodes into three pieces: a 20 g piece with velocity 25 î m/s, a 15 g piece with velocity −10 î + 10ĵ m/s, and a 10 g piece with velocity −15 î − 20ĵ m/s. What was the projectile's velocity just before the explosion?

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A 2100 kg truck is traveling east through an intersection at 2.0 m/s when it is hit simultaneously from the side and the rear. (Some people have all the luck!) One car is a 1200 kg compact traveling north at 5.0 m/s. The other is a 1500 kg midsize traveling east at 10 m/s. The three vehicles become entangled and slide as one body. What are their speed and direction just after the collision?

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The nucleus of the polonium isotope ²¹⁴Po (mass 214 u) is radioactive and decays by emitting an alpha particle (a helium nucleus with mass 4 u). Laboratory experiments measure the speed of the alpha particle to be 1.92×10⁷ m/s . Assuming the polonium nucleus was initially at rest, what is the recoil speed of the nucleus that remains after the decay?

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