Velocity is given by

$\overline{){\mathbf{v}}{\mathbf{=}}{\mathbf{a}}{\mathbf{t}}}$

Newton's second law:

$\overline{){\mathbf{\Sigma}}{\mathbf{F}}{\mathbf{=}}{\mathbf{m}}{\mathbf{a}}}$

**(a)**

Solving for a, from velocity and Newton's second law:

a = **v/t = F/m**

v = (F/m)t

Using the expression for momentum:

**Learning Goal:**

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*? *d**t*, (1)

where ∑*F*? is the *net* force *F*? net acting on the object, and *d**p*? *d**t* is the rate at which the object's momentum is changing.

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

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

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

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

This equation is known as the *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* *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)

Here *F*, *v*1, and *v*2 are the *components* 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 the *x* and *y* 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 mass *m* moving along the *x* axis. The net force *F* is acting on the particle along the *x* axis. *F* is a constant force.)

A) The particle starts from rest at *t*=0. What is the magnitude *p* of the momentum of the particle at time *t*? Assume that *t*>0.

B) The particle starts from rest at *t*=0. What is the magnitude *v* of the velocity of the particle at time *t*? Assume that *t*>0.

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

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

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