Problem 11 - University Physics with Modern Physics

Question

CALC At time \(t=0,\) a 2150 -kg rocket in outer space fires an engine that exerts an increasing force on it in the \(+x\) -direction. This force obeys the equation \(F_{x}=A t^{2},\) where \(t\) is time, and has a magnitude of 781.25 \(\mathrm{N}\) when \(t=1.25 \mathrm{s}\) . (a) Find the SI value of the constant \(A,\) including its units. (b) What impulse does the engine exert on the rocket during the 1.50 -s interval starting 2.00 \(\mathrm{s}\) after the engine is fired? (c) By how much does the rocket's velocity change during this interval?

Step-by-Step Solution

Answer

(a) A = 500.0 N/s^2. (b) Impulse = 5812.5 Ns. (c) Velocity change = 2.703 m/s.

Step 1: Determine Constant A

Given \(F_x = A t^2\) and \(F_x = 781.25 \, \text{N}\) at \(t = 1.25 \, \text{s}\), substitute into the formula:\[781.25 = A (1.25)^2\]Solve for \(A\):\[A = \frac{781.25}{1.25^2} = \frac{781.25}{1.5625} = 500.0 \, \text{N/s}^2\]Thus, the constant \(A = 500.0 \, \text{N/s}^2\).

Step 2: Calculate the Impulse

Impulse \(J\) is the integral of force over time:\[J = \int_{t_1}^{t_2} F(t) \, dt = \int_{2.00}^{3.50} A t^2 \, dt\]Substitute \(A = 500.0\):\[J = \int_{2.00}^{3.50} 500 t^2 \, dt\]Compute the integral:\[J = 500 \left[ \frac{t^3}{3} \right]_{2.00}^{3.50} = 500 \left( \frac{3.50^3}{3} - \frac{2.00^3}{3} \right)\]\[J = 500 \left( \frac{42.875}{3} - \frac{8.000}{3} \right) = 500 \left( 14.2917 - 2.6667 \right)\]\[J = 500 \times 11.625 = 5812.5 \, \text{Ns}\]

Step 3: Calculate Change in Velocity

The change in velocity \(\Delta v\) can be found using the relationship \(J = m \Delta v\):\[\Delta v = \frac{J}{m} = \frac{5812.5}{2150} \]\[\Delta v = 2.703 \, \text{m/s}\]

Key Concepts

Newton's Second Law of Motion

Newton's Second Law of Motion is a fundamental principle of classical physics, guiding our understanding of how forces affect the motion of an object. Simply put, this law states that an object accelerates when a force is applied to it. The relationship can be expressed mathematically as \[ F = ma \].
Here, \( F \) represents the force applied, \( m \) stands for the object's mass, and \( a \) is the acceleration produced.

In the context of rocket propulsion, this means that the engine's force is directly related to the change in motion of the rocket.
For example, if a 2150-kg rocket experiences an increasing force as described by \( F_x = At^2 \), this force dictates the acceleration the rocket undergoes.
Since the force depends on time, the acceleration and thus the motion of the rocket will also vary with time.

• The bigger the force, the greater the acceleration.
• An increase in mass requires more force to achieve the same acceleration.

Consequently, understanding Newton's Second Law is crucial for determining how changes in force impact the motion of rockets and other objects.

Impulse-Momentum Theorem

The Impulse-Momentum Theorem bridges the concept of impulse, the product of force and time, with momentum changes in an object. It's represented by the equation: \[ J = ext{Impulse} = F imes ext{time} = ext{Change in Momentum} \]
Impulse is essentially the "oomph" that a force gives to an object over a period of time.
In simpler terms, when a force acts on an object for a certain time, it changes the object's momentum.
Momentum, denoted by \( p \), can be defined as \( p = mv \), where \( m \) is mass and \( v \) is velocity.
Therefore, a change in momentum implies a velocity change:
\[ J = riangle p = m imes riangle v \]

• Greater impulse results in a larger momentum change.
• A prolonged application of force (long time) can lead to significant velocity changes.

In the rocket scenario, the impulse delivered by the engine over 1.5 seconds changes the rocket's momentum, measured by how much the velocity is altered."

Calculus in Physics

Calculus plays an essential role in understanding the dynamics of physics, where quantities change continuously over time. Concepts such as differentiation and integration allow us to describe and predict these changes accurately.

Integration in Calculus

When we talk about forces that change over time, like the rocket's force given by \( F_x = At^2 \), calculus helps us determine total effects such as impulse.
The impulse can be calculated as the integral of force with respect to time, i.e., \[ J = \int F(t) \, dt \]This calculates the total impulse applied by the varying force over a specified time interval.

• This integration tells us how changes in force accumulate over time.
• It reveals the overall impact of time-varying forces on an object's motion.

Finding Velocity

Similarly, calculus allows us to find velocity changes when the force is not constant. By integrating force over time and knowing the object's mass, we can determine how velocity changes.

• The interplay of force, time, and mass is crucial for predicting real-world motion effectively.

In essence, calculus provides the tools necessary to tackle complex physics problems involving variability and time-dependence.