On a test flight a rocket with mass 400 kg blasts off from the surface of the earth. The rocket

On a test flight a rocket with mass 400 kg blasts off from...

On a test flight a rocket with mass $400 \mathrm{~kg}$ blasts off from the surface of the earth. The rocket engines apply a constant upward force $F$ until the rocket reaches a height of $100 \mathrm{~m}$ and then they shut off. If the rocket is to reach a maximum height of $400 \mathrm{~m}$ above the surface of the earth, what value of $F$ is required? Assume the change in the rocket's mass is negligible.

Key Concepts

Kinematics under Constant Acceleration

Kinematics under Constant Acceleration involves the equations that describe the motion of objects when acceleration is constant. These equations link variables such as displacement, initial velocity, acceleration, and time. They are essential for calculating the motion of a rocket as it transitions from powered flight to a free-fall trajectory under gravity until it reaches its maximum height.

Gravitational Potential Energy

Gravitational Potential Energy is the energy an object possesses due to its position in a gravitational field, quantified as the product of the object's mass, gravitational acceleration, and height above a reference point. This concept is key in determining the energy required for a rocket to reach a certain altitude, as the additional energy must overcome the increase in potential energy associated with gaining height.

Work-Energy Principle

The Work-Energy Principle explains that the work done by forces on an object results in a change in its kinetic energy. In problems involving a force applied over a distance, like a rocket's engine providing thrust over a specific altitude, this principle is used to determine the energy imparted to the object, which then converts into kinetic energy during powered flight and eventually into gravitational potential energy during free flight.

Newton's Second Law

Newton's Second Law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. This concept is crucial for determining how much additional force, beyond gravity, is needed to accelerate an object—like a rocket—in order to change its motion or overcome gravitational pull.

Transcript

00:02 Hi, in the given problem, the test rocket is going up, having a constant force exerted on it by its engine.

00:17 Suppose that constant force acting on the rocket is f, under which it moves through 100 meters.

00:40 So for this accelerated motion of the rocket for first 100 meter, its weight m .g will be acting vertically downward.

01:15 The force, constant force, is acting on it vertically upward.

01:20 So the net force acting on it will be f minus mg.

01:25 That will be kept equal to m .a.

01:29 As per newton's second law of motion.

01:32 So acceleration in the motion of rocket in upward direction will be given by f minus mg divided by m, means mass of the rocket.

01:45 Here, this mass is given as 400 kilogram using this acceleration and third equation of motion, which says, vf squared square of final velocity achieved by an object is equal to v i squared the initial velocity square of initial velocity plus 2a into s the distance moved as the rocket was initially at rest so vi will be put 0 hence here we get vf square is equal to 0 plus 2 into f minus m g divided by m into for s means this is height raised by the rocket which is given as 100 meter.

02:51 So this vf squared comes out to be 2 into 100 means 200 f minus m g divided by m hence this vf will come out to be square root of 200 f minus m g by m we are using it just as an expression so we are not putting the value of mass or acceleration due to gravity force is missing here we have to find it now for the further motion of the rocket when the engine is not working the acceleration will be acceleration to gravity and that will be acting vertically downward.

03:43 So now for the further motion of rocket under acceleration due to gravity, the final velocity achieved by the rocket under the force will become initial velocity of the rocket for this retarded motion...

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