Ch 11: Impulse and Momentum

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 11: Impulse and Momentum

문제 70

11장, 문제 70

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?

검증된 단계별 안내

Step 1: Understand the principle of conservation of momentum. In the absence of external forces, the total momentum of a system before and after an event (like an explosion) remains constant. Here, the total momentum of the projectile before the explosion equals the total momentum of its fragments after the explosion.

Step 2: Write the expression for the total momentum of the system before the explosion. Let the velocity of the projectile before the explosion be $ \vec{v}_i $. The mass of the projectile is $ m = 45 \ \text{g} = 0.045 \ \text{kg} $. The total momentum before the explosion is $ \vec{p}_i = m \vec{v}_i $.

Step 3: Write the expression for the total momentum of the system after the explosion. The momentum of each fragment is the product of its mass and velocity. For the three fragments, the total momentum after the explosion is: $ \vec{p}_f = m_1 \vec{v}_1 + m_2 \vec{v}_2 + m_3 \vec{v}_3 $, where $ m_1 = 0.020 \ \text{kg}, \ m_2 = 0.015 \ \text{kg}, \ m_3 = 0.010 \ \text{kg} $, and their respective velocities are given.

Step 4: Substitute the given velocities into the expression for $ \vec{p}_f $. Using the velocities $ \vec{v}_1 = 25 \hat{i} \ \text{m/s}, \ \vec{v}_2 = -10 \hat{i} + 10 \hat{j} \ \text{m/s}, \ \vec{v}_3 = -15 \hat{i} - 20 \hat{j} \ \text{m/s} $, calculate the total momentum after the explosion: $ \vec{p}_f = (0.020)(25 \hat{i}) + (0.015)(-10 \hat{i} + 10 \hat{j}) + (0.010)(-15 \hat{i} - 20 \hat{j}) $.

Step 5: Set $ \vec{p}_i = \vec{p}_f $ to solve for $ \vec{v}_i $. Since $ \vec{p}_i = m \vec{v}_i $, rearrange to find $ \vec{v}_i = \frac{\vec{p}_f}{m} $. Substitute the total momentum $ \vec{p}_f $ and the mass $ m = 0.045 \ \text{kg} $ to determine the initial velocity of the projectile.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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. In this case, the momentum of the projectile before it explodes must equal the combined momentum of the three pieces after the explosion, allowing us to calculate the initial velocity of the projectile.

Momentum Calculation

Momentum is calculated as the product of mass and velocity (p = mv). For each piece of the projectile after the explosion, we can calculate its momentum by multiplying its mass by its velocity vector. Summing these momenta will give us the total momentum after the explosion, which we can then set equal to the initial momentum to find the projectile's velocity.

Vector Addition

Since momentum is a vector quantity, we must consider both the magnitude and direction of the velocities involved. Vector addition is used to combine the momentum vectors of the pieces after the explosion. This involves adding the components of the vectors separately in the x and y directions to find the resultant momentum vector, which is essential for determining the initial velocity of the projectile.

관련 실천

교과서 질문

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).

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

793

views

교과서 질문

Consider a partially elastic collision in which ball A of mass m with initial velocity (v_ix)_A collides with stationary ball B, also of mass m, and in which 1/4 of the mechanical energy is dissipated as thermal energy. Find expressions for the final velocities of each ball. Hint: Mathematically there are two solutions; however, one of them is physically impossible.

1754

views

교과서 질문

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?

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?

1765

views

교과서 질문

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?

2201

views

교과서 질문

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?

2150

views