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 73

Chapter 11, Problem 73

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?

Verified step by step guidance

Step 1: Recognize that this is a perfectly inelastic collision problem where the three vehicles stick together after the collision. The principle of conservation of momentum applies separately in the x-direction (east-west) and the y-direction (north-south).

Step 2: Write the momentum conservation equation for the x-direction. The total momentum in the x-direction before the collision is the sum of the momenta of the truck and the midsize car (since the compact car has no velocity in the x-direction). Use the formula for momentum: $ p = m \cdot v $, where $ m $ is mass and $ v $ is velocity.

Step 3: Write the momentum conservation equation for the y-direction. The total momentum in the y-direction before the collision is the momentum of the compact car (since the truck and midsize car have no velocity in the y-direction).

Step 4: Calculate the total mass of the entangled vehicles after the collision. The total mass is the sum of the masses of the truck, compact car, and midsize car. Use this total mass to find the final velocity components $ v_x $ (in the x-direction) and $ v_y $ (in the y-direction) by dividing the total momentum in each direction by the total mass.

Step 5: Determine the magnitude and direction of the final velocity. The magnitude of the velocity is given by $ v = \sqrt{v_x^2 + v_y^2} $, and the direction (angle $ \theta $) is given by $ \theta = \tan^{-1}(v_y / v_x) $.

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Key Concepts

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

Conservation of Momentum

The principle of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it. In collisions, the momentum before the collision must equal the momentum after the collision. This concept is crucial for solving problems involving collisions, as it allows us to calculate the final velocities of the objects involved.

Vector Addition

In physics, velocity is a vector quantity, meaning it has both magnitude and direction. When multiple objects collide, their velocities must be treated as vectors, requiring vector addition to determine the resultant velocity of the combined mass. This involves breaking down the velocities into their components and summing them accordingly to find the overall speed and direction after the collision.

Inelastic Collision

An inelastic collision is a type of collision where the colliding objects stick together after impact, resulting in a loss of kinetic energy. In this scenario, the three vehicles become entangled and move as a single body, which is characteristic of inelastic collisions. Understanding this concept is essential for analyzing the final speed and direction of the combined mass post-collision.

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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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Textbook Question

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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Textbook Question

A spaceship of mass 2.0×10⁶ kg is cruising at a speed of 5.0×10⁶ m/s when the antimatter reactor fails, blowing the ship into three pieces. One section, having a mass of 5.0×10⁵ kg, is blown straight backward with a speed of 2.0×10⁶ m/s . A second piece, with mass 8.0×10⁵ kg, continues forward at 1.0×10⁶ m/s. What are the direction and speed of the third piece?

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Textbook Question

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?

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Textbook Question

A neutron is an electrically neutral subatomic particle with a mass just slightly greater than that of a proton. A free neutron is radioactive and decays after a few minutes into other subatomic particles. In one experiment, a neutron at rest was observed to decay into a proton (mass 1.67×10^-27 kg) and an electron (mass 9.11×10^-31 kg) . The proton and electron were shot out back-to-back. The proton speed was measured to be 1.0 ×10⁵ m/s, and the electron speed was 3.0×10⁷ m/s. No other decay products were detected. How much momentum did this neutrino 'carry away' with it?

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