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 61

Chapter 11, Problem 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?

Verified step by step guidance

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

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.

Recommended video:

Guided course

08:25

Kinematics Equations

Related Practice

Textbook Question

Two 500 g blocks of wood are 2.0 m apart on a frictionless table. A 10 g bullet is fired at 400 m/s toward the blocks. It passes all the way through the first block, then embeds itself in the second block. The speed of the first block immediately afterward is 6.0 m/s . What is the speed of the second block after the bullet stops in it?

views

Textbook Question

A 100 g ball moving to the right at 4.0 m/s collides head-on with a 200g ball that is moving to the left at 3.0 m/s. If the collision is perfectly elastic, what are the speed and direction of each ball after the collision?

views

Textbook Question

The stoplight had just changed and a 2000 kg Cadillac had entered the intersection, heading north at 3.0 m/s , when it was struck by a 1000 kg eastbound Volkswagen. The cars stuck together and slid to a halt, leaving skid marks angled 35° north of east. How fast was the Volkswagen going just before the impact?

views

Textbook Question

A 1500 kg weather rocket accelerates upward at 10 m/s². It explodes 2.0 s after liftoff and breaks into two fragments, one twice as massive as the other. Photos reveal that the lighter fragment traveled straight up and reached a maximum height of 530 m. What were the speed and direction of the heavier fragment just after the explosion?

views

Textbook Question

A 30 ton rail car and a 90 ton rail car, initially at rest, are connected together with a giant but massless compressed spring between them. When released, the 30 ton car is pushed away at a speed of 4.0 m/s relative to the 90 ton car. What is the speed of the 30 ton car relative to the ground?

views

Textbook Question

One end of a massless, 30-cm-long spring with spring constant 15 N/m is attached to a 250 g stationary air-track glider; the other end is attached to the track. A 500 g glider hits and sticks to the 250 g glider, compressing the spring to a minimum length of 22 cm. What was the speed of the 500 g glider just before impact?

views

A 500 g particle has velocity vx = −5.0 m/s at t = −2 s. Force Fx = (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?

Verified video answer for a similar problem:

Key Concepts

Newton's Second Law of Motion

Impulse and Momentum

Kinematics

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?