91Ó°ÊÓ

Newton's Second Law of Motion is fundamental in understanding how forces affect the motion of objects. It states that the force acting on an object is equal to the mass of that object multiplied by its acceleration, mathematically represented as \( F = ma \).
This law helps us determine how a force will change the motion of an object. In the case of the blimp in our problem, a force \( F = kt \) is applied. Here, \( k \) is a constant and \( t \) is time.
The law tells us that the force results in an acceleration of the blimp. Since the blimp starts at rest, we first solve \( F = ma \) to find the acceleration \( a = \frac{F}{m} = \frac{kt}{m} \).
The beauty of Newton's Second Law is its ability to connect dynamics—how a force leads to acceleration—with the kinematic equations, which describe motion.

Integration is a mathematical technique used to find quantities like area under a curve, or to go from a rate of change (like acceleration) to an accumulated total (like velocity). In this problem, we use integration to find both velocity and position from acceleration.
First, we integrate acceleration to find velocity. We have \( a = \frac{kt}{m} \), so we integrate this expression with respect to time \( \int a(t) \, dt \). This gives us the velocity function \( v(t) = \frac{k}{2m} t^2 \), taking into account that the initial velocity is zero.
Next, to find the blimp's position as a function of time, we integrate the velocity function \( v(t) \).

• Velocity \( v(t) \) represents the rate of change of the position.
• By integrating \( v(t) = \frac{k}{2m} t^2 \), we find the position equation \( x(t) = \frac{k}{6m} t^3 \).

The process of integration is crucial here as it lets us transition from knowing how fast things change to knowing the state of motion.

Acceleration is the rate of change of velocity with respect to time. It's a vector quantity, which means it has both magnitude and direction. In scenarios governed by linear motion—like the blimp—acceleration tells us how quickly velocity is increasing or decreasing.
In this problem, the blimp's acceleration comes from the force exerted by the propeller, described by \( a = \frac{F}{m} = \frac{kt}{m} \).

• As time progresses, the force—and hence the acceleration—increases linearly since \( F = kt \).
• This results in a quadratic increase in velocity because velocity is the integral of acceleration.

Essentially, the steady increase in force ensures that the acceleration also steadily increases over time, gradually speeding up the blimp.

Velocity indicates how fast something is moving in a specific direction. It's different from speed since it includes direction. In our exercise, once we know the acceleration \( a = \frac{kt}{m} \), we can find the velocity by integrating the acceleration over time.
The blimp starts from rest, so its initial velocity is zero. We find the velocity function by the integral:
\[ v(t) = \int a(t) \, dt = \frac{k}{2m} t^2 \]
This shows that velocity is not just a simple function of time \( t \), but of time squared \( t^2 \).

• This quadratic nature indicates that as time goes on, the blimp gains speed more rapidly.
• The blimp starts slowly, but its velocity increases as \( t \) increases.

This insight into velocity helps us understand how objects accelerate and gain speed over time due to an applied force.