which of the vectors below best represents the direction of the impulse vector jā ?

Ā

To learn about the impulse-momentum theorem and its applications in some common cases.

Using the concept of momentum, Newton's second law can be rewritten as

?*F*? =*d**p*?Ā *d**t*, (1)

where ?*F*?Ā Ā is theĀ *net*Ā forceĀ *F*? net acting on the object, andĀ *d**p*?Ā *d**t*Ā is the rate at which the object's momentum is changing.

If the object is observed during an interval of time between timesĀ *t*1 andĀ *t*2, then integration of both sides of equation (1) gives

?*t*2*t*1?*F*?Ā *d**t*=?*t*2*t*1*d**p*?Ā *d**t**d**t*. (2)

The right side of equation (2) is simply the change in the object's momentumĀ *p*2??*p*1?. The left side is called theĀ *impulse of the net force*Ā and is denoted byĀ *J*? . Then equation (2) can be rewritten as

*J*? =*p*2??*p*1?.

This equation is known as theĀ *impulse-momentum theorem*. It states that the change in an object's momentum is equal to the impulse of the net force acting on the object. In the case of a constant *net force*Ā *F*? net acting along the direction of motion, the impulse-momentum theorem can be written as

*F*(*t*2?*t*1)=*m**v*2?*m**v*1. (3)

HereĀ *F*,Ā *v*1, andĀ *v*2 are theĀ *components*Ā of the corresponding vector quantities along the chosen coordinate axis. If the motion in question is two-dimensional, it is often useful to apply equation (3) to theĀ *x*Ā andĀ *y*Ā components of motion separately.

( The following questions will help you learn to apply the impulse-momentum theorem to the cases of constant and varying force acting along the direction of motion. First, let us consider a particle of massĀ *m*Ā moving along the *x-axis*. The net forceĀ *F*Ā is acting on the particle along the *x-axis*.Ā *F*Ā is a constant force.)

A) The particle starts from rest atĀ *t*=0. What is the magnitudeĀ *p*Ā of the momentum of the particle at timeĀ *t*? Assume thatĀ *t*>0.

B)Ā The particle starts from rest atĀ *t*=0. What is the magnitudeĀ *v*Ā of the velocity of the particle at timeĀ *t*? Assume thatĀ *t*>0.

C) The particle has the momentum of magnitude *p*1 at a certain instant. What isĀ *p*2, the magnitude of its momentum ?*t*Ā seconds later?

D) The particle has the momentum of magnitude *p*1 at a certain instant. What isĀ *v*2, the magnitude of its velocity ?*t*Ā seconds later?

E)Ā Find the magnitude of the impulseĀ *J*Ā delivered to the particle.

F) Which of the vectors below best represents the direction of the impulse vectorĀ *J*??

(Figure 1)

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