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 61

11장, 문제 61

A 500 g particle has velocity v_x = −5.0 m/s at t = −2 s. Force F_x= (4−t²) N, where t is in s, is exerted on the particle between t = −2 s and t = 2 s. This force increases from 0 N at t = −2 s to 4 N at t = 0 s and then back to 0 N at t = 2 s. What is the particle's velocity at t = 2s?

검증된 단계별 안내

Step 1: Identify the relationship between force and acceleration using Newton's second law. The formula is:

F = m a

, where

F

is the force,

m

is the mass, and

a

is the acceleration. Rearrange to find acceleration:

a = \frac{F}{m}

Step 2: Express the acceleration as a function of time. The force is given as

F = 4 - t^{2}

, and the mass of the particle is

0.5

kg. Substitute these values into the formula for acceleration:

a = \frac{4 - t^{2}}{0.5}

Step 3: Use the relationship between acceleration and velocity. Acceleration is the derivative of velocity with respect to time:

a = \frac{d v}{d}

. Rearrange to express velocity as an integral of acceleration:

v = v_{0} + \int_{t = - 2}^{t = 2} a d t

, where

v_{0}

is the initial velocity.

Step 4: Substitute the expression for acceleration into the integral. Replace

a

with

\frac{4 - t^{2}}{0.5}

in the integral:

v = v_{0} + \int_{t = - 2}^{t = 2} \frac{4 - t^{2}}{0.5} d t

Step 5: Solve the integral to find the velocity at

t = 2

. Evaluate the definite integral of the acceleration function over the interval

[- 2, 2]

, and add the result to the initial velocity

v_{0} = - 5

m/s to find the final velocity.

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주요 개념

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

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is expressed mathematically as F = ma, where F is the net force, m is the mass, and a is the acceleration. Understanding this law is crucial for analyzing how the force applied to the particle affects its motion over time.

Impulse and Momentum

Impulse is defined as the change in momentum of an object when a force is applied over a period of time. It is calculated as the product of the average force and the time duration during which the force acts. The relationship between impulse and momentum is given by the equation Impulse = Δp = F_avg * Δt, where Δp is the change in momentum. This concept is essential for determining how the force affects the particle's velocity from t = -2 s to t = 2 s.

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration. In this problem, kinematic equations can be used to relate the initial velocity, the change in velocity due to the applied force, and the final velocity of the particle at t = 2 s.