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Chapter 4: Problem 47

A small rocket with mass \(20.0 \mathrm{~kg}\) is moving in free fall toward the earth. Air resistance can be neglected. When the rocket is \(80.0 \mathrm{~m}\) above the surface of the earth, it is moving downward with a speed of \(30.0 \mathrm{~m} / \mathrm{s}\). At that instant the rocket engines start to fire and produce a constant upward force \(F\) on the rocket. Assume the change in the rocket's mass is negligible. What is the value of \(F\) if the rocket's speed becomes zero just as it reaches the surface of the earth, for a soft landing? (Hint: The net force on the rocket is the combination of the upward force \(F\) from the engines and the downward weight of the rocket.)

Short Answer

Expert verified
The value of \(F\), the constant upward force required for the rocket to stop before it hits the earth surface, can be found by solving the equations formed from Newton's second law of motion and the kinematic equation.

Step by step solution

01

Understand the Forces involved

Since there is no air resistance, the only downward force on the rocket is its weight given by \(F_g = mg\), where \(m\) is the mass of the rocket and \(g\) is the acceleration due to gravity. Since it's not specified, we'll assume \(g = 9.81 m/s虏\). Thus, \(F_g = 20 kg * 9.81 m/s虏 = 196.2 N\). The net force on the rocket at any moment consists of both the force produced by the engines \(F\) and the weight of the rocket itself.
02

Apply Newton's second law

Applying Newton's second law of motion (\(F_{net}=ma\)), where \(F_{net}\) is the net force on the rocket, \(m\) is its mass and \(a\) its acceleration. In this case, since the rocket is stopping we can say the final speed is 0 m/s and thus the acceleration \(a = (v_f - v_i)/t = (0 - 30 m/s)/t\). Applying \(F_{net}=ma\) we get \(F_{net} = m*(0 - v_i)/t = -mv_i/t\). Since the net force \(F_{net}\) is combination of downward and upward forces, we could express it as \(F_{net} = F - F_g\) where \(F_g\) is the weight of the rocket and \(F\) is the upward rocket force.
03

Solve for Force F

Solving the net force equation for \(F\) we get, \(F = F_{net} + F_g = -mv_i/t + F_g\). The time \(t\) to stop the rocket is given by kinematic equation \(d = v_it + \frac{1}{2}a*t虏\). Since \(d = 80m\), \(v_i = 30m/s\) and \(a = (0 - v_i)/t\), we can solve this equation to find \(t\). After finding the value of \(t\), we will substitute its value into the force equation to find the force \(F\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rocket Motion
Rocket motion involves understanding how forces affect the rocket's movement. When engines fire, they create a thrust that propels the rocket. In the exercise, the engines provide an upward force that counters the rocket's downward motion due to gravity. It's crucial to remember that even in high-speed movements, such as a rocket in motion, it follows Newton's laws of motion. Here, we're concerned with Newton's second law which connects the force, mass, and acceleration of the rocket.
  • The motion begins with the rocket descending under gravity.
  • Once the engines ignite, the thrust must overcome its current motion for a soft landing.
  • The applied force reduces the rocket's descent to zero as it reaches the ground.
Understanding these forces is key to predicting how and when the rocket will stop, and how much engine power is needed to achieve that soft landing.
Kinematics
Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause the motion. In the context of the exercise, we use kinematic equations to predict the movement of the rocket and calculate critical variables like time and distance.
The main focus here is using the equation \[ v^2 = u^2 + 2as \] where:
  • \(v\) is the final velocity, here 0 m/s for a soft landing,
  • \(u\) is the initial downward velocity (30 m/s),
  • \(a\) is the acceleration caused by the net force, and
  • \(s\) is the distance (80 m).
Using these formulas, we determine how quickly the rocket will slow down after the force is applied, and this helps find the exact time it takes to land the rocket softly. It all revolves around how initial conditions are transformed by kinematic rules until the rocket touches down safely.
Free Fall
Free fall describes the condition when gravity is the only force acting on an object. Initially, the rocket experiences free fall toward the Earth before the engines are activated. Prior to engine ignition, its motion complies with uniform acceleration due to gravity.
In this scenario:
  • The initial velocity is directed downward at 30 m/s.
  • Gravity accelerates the rocket further at 9.81 m/s虏.
  • Free fall continues until the engines create an opposing force.
Understanding free fall is essential to calculate the changes in the rocket's motion once the opposing engine force is applied. This comprehension provides insight into how the force neutralizes gravity's effect, enabling the rocket to achieve a smooth landing.
Forces
Forces play a pivotal role in the rocket's journey from descent to landing. In the exercise, we must find the value of the upward force that stops the downward motion. Newton's second law states that \[ F_{net} = ma \] and guides our calculation. Here, the net force is the total of thrust minus the weight of the rocket.
The weight is calculated as:
  • \( F_g = mg \) where \( m \) is the mass and \( g \) is gravitational acceleration, resulting in 196.2 N.
To find the thrust \( F \), we consider:
  • The force needs to not only balance the downward pull of gravity but also bring the initial downward velocity to zero.
  • We first determine the total time and acceleration required.
  • The final calculation combines these factors to pinpoint the necessary thrust.
By understanding these forces, you can predict and control the rocket's motion to ensure a successful mission.

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Most popular questions from this chapter

Two crates, one with mass \(4.00 \mathrm{~kg}\) and the other with mass \(6.00 \mathrm{~kg},\) sit on the frictionless surface of a frozen pond, connected by a light rope (Fig. P4.37). A woman wearing golf shoes (for traction) pulls horizontally on the \(6.00 \mathrm{~kg}\) crate with a force \(F\) that gives the crate an acceleration of \(2.50 \mathrm{~m} / \mathrm{s}^{2}\). (a) What is the acceleration of the \(4.00 \mathrm{~kg}\) crate? (b) Draw a free-body diagram for the \(4.00 \mathrm{~kg}\) crate. Use that diagram and Newton's second law to find the tension \(T\) in the rope that connects the two crates. (c) Draw a free-body diagram for the \(6.00 \mathrm{~kg}\) crate. What is the direction of the net force on the \(6.00 \mathrm{~kg}\) crate? Which is larger in magnitude, \(T\) or \(F ?\) (d) Use part (c) and Newton's second law to calculate the magnitude of \(F\).

Forces \(\vec{F}_{1}\) and \(\vec{F}_{2}\) act at a point. The magnitude of \(\vec{F}_{1}\) is \(9.00 \mathrm{~N}\), and its direction is \(60.0^{\circ}\) above the \(x\) -axis in the second quadrant. The magnitude of \(\vec{F}_{2}\) is \(6.00 \mathrm{~N},\) and its direction is \(53.1^{\circ}\) below the \(x\) -axis in the third quadrant. (a) What are the \(x\) - and \(y\) -components of the resultant force? (b) What is the magnitude of the resultant force?

Boxes \(A\) and \(B\) are in contact on a horizontal, frictionless surface (Fig. E4.23). Box A has mass \(20.0 \mathrm{~kg}\) and box \(B\) has mass \(5.0 \mathrm{~kg} .\) A horizontal force of \(250 \mathrm{~N}\) is exerted on box \(A\). What is the magnitude of the force that box \(A\) exerts on box \(B ?\)

You have just landed on Planet X. You release a \(100 \mathrm{~g}\) ball from rest from a height of \(10.0 \mathrm{~m}\) and measure that it takes \(3.40 \mathrm{~s}\) to reach the ground. Ignore any force on the ball from the atmosphere of the planet. How much does the \(100 \mathrm{~g}\) ball weigh on the surface of Planet X?

Estimate the mass in kilograms and the weight in pounds of a typical sumo wrestler. How do your estimates for the wrestler compare to your estimates of the average mass and weight of the students in your physics class? Do a web search if necessary to help make the estimates. In your solution list what values you assume for the quantities you use in making your estimates.
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