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? =dp?Ā dt, (1)
where ?F?Ā Ā is theĀ netĀ forceĀ F? net acting on the object, andĀ dp?Ā dtĀ is the rate at which the object's momentum is changing.
If the object is observed during an interval of time between timesĀ t1 andĀ t2, then integration of both sides of equation (1) gives
?t2t1?F?Ā dt=?t2t1dp?Ā dtdt. (2)
The right side of equation (2) is simply the change in the object's momentumĀ p2??p1?. The left side is called theĀ impulse of the net forceĀ and is denoted byĀ J? . Then equation (2) can be rewritten as
J? =p2??p1?.
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 forceĀ F? net acting along the direction of motion, the impulse-momentum theorem can be written as
F(t2?t1)=mv2?mv1. (3)
HereĀ F,Ā v1, andĀ v2 are theĀ componentsĀ 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 xĀ andĀ yĀ 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 mĀ moving along the x-axis. The net forceĀ FĀ is acting on the particle along the x-axis.Ā FĀ is a constant force.)
A) The particle starts from rest atĀ t=0. What is the magnitudeĀ pĀ 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 vĀ of the velocity of the particle at timeĀ t? Assume thatĀ t>0.
C) The particle has the momentum of magnitude p1 at a certain instant. What isĀ p2, the magnitude of its momentum ?tĀ seconds later?
D) The particle has the momentum of magnitude p1 at a certain instant. What isĀ v2, the magnitude of its velocity ?tĀ seconds later?
E)Ā Find the magnitude of the impulseĀ JĀ 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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