To determine the correct equation for the impulse applied by a force, let's review the concept of impulse in physics. Impulse (denoted by [tex]\( I \)[/tex]) is a measure of the change in momentum of an object when it is acted upon by a force over a given time interval. The formula for impulse is derived from Newton's second law of motion.
Newton's second law states that the force [tex]\( F \)[/tex] applied to an object is equal to the rate of change of its momentum [tex]\( p \)[/tex]:
[tex]\[ F = \frac{dp}{dt} \][/tex]
Impulse can be calculated by integrating force over the time interval [tex]\( \Delta t \)[/tex]:
[tex]\[ I = \int_{t_1}^{t_2} F \, dt \][/tex]
For a constant force [tex]\( F \)[/tex] acting over a time interval [tex]\( \Delta t \)[/tex], this simplifies to:
[tex]\[ I = F \Delta t \][/tex]
Now, let's analyze the given options:
A. [tex]\( I = F \Delta t \)[/tex] \\
This matches our understanding of impulse as the product of force and the time interval over which it acts. This is the correct equation for impulse.
B. [tex]\( I = \frac{p}{m} \)[/tex] \\
This equation suggests dividing momentum [tex]\( p \)[/tex] by mass [tex]\( m \)[/tex], which gives velocity rather than impulse. This is incorrect.
C. [tex]\( I = \frac{1}{2} k x^2 \)[/tex] \\
This formula represents the potential energy stored in a spring, according to Hooke's law, where [tex]\( k \)[/tex] is the spring constant and [tex]\( x \)[/tex] is the displacement. This is irrelevant to impulse.
D. [tex]\( 1 = -k x \)[/tex] \\
This equation seems to be a form of Hooke's law for a spring, where the force [tex]\( F \)[/tex] is directly proportional to the displacement [tex]\( x \)[/tex] (and [tex]\( k \)[/tex] is the spring constant). It does not represent impulse.
Given our understanding of impulse, the correct equation is:
A. [tex]\( I = F \Delta t \)[/tex]
