A small rocket-propelled vehicle weighs 125 lb, in cluding 20 lb of fuel. Fuel is burned at the

A small rocket-propelled vehicle weighs 125 lb, in cluding...

A small rocket-propelled vehicle weighs $125 \mathrm{lb}$, in cluding 20 lb of fuel. Fuel is burned at the constant rate of 2 lb/sec with an exhaust velocity relative to the nozzle of $400 \mathrm{ft} / \mathrm{sec}$. Upon ignition the vehicle is released from rest on the $10^{\circ}$ incline. Calculate the maximum velocity $v$ reached by the vehicle. Neglect all friction.

Key Concepts

Rocket Propulsion

This concept deals with the dynamics of vehicles that produce thrust by expelling mass at high velocity. It involves understanding how the expulsion of fuel (mass) generates a reactive force (thrust) that propels the vehicle in the opposite direction according to Newton’s third law.

Conservation of Momentum

A fundamental principle applied to systems where mass changes over time. In the context of rockets, conservation of momentum is used to relate the momentum lost by the expelled fuel to the momentum gained by the rocket, forming the basis for equations that describe the vehicle's acceleration.

Variable Mass Systems

Unlike constant-mass systems, rockets lose mass as fuel burns. This requires the use of differential equations in which both the vehicle's mass and velocity change with time, complicating the dynamics compared to fixed-mass objects.

Inclined Plane Dynamics

When a vehicle moves along a slope, gravitational forces must be resolved into components. For an incline, the component of gravity parallel to the surface (found using the sine of the incline angle) opposes the motion and must be considered when analyzing the net force and acceleration.

Thrust Force Calculation

This involves computing the force generated by the rocket's engine, which is the product of the exhaust velocity (relative to the nozzle) and the rate at which mass (fuel) is expelled. This force drives the rocket forward and is counteracted by gravitational forces, particularly on an incline.

Transcript

00:02 So here on this problem, we are given this small rocket that is released from rest on a 10 degree inclined plane as you can see here.

00:08 Now this rocket has a total weight of 125 pound and initial weight for the fuel in the rocket.

00:20 This value we are given to be as 20 pounds.

00:23 This is the initial fuel and the fuel is burned at a constant rate of two pound per second and the relative exhaust velocity we are given to be as 400 feet per second now we have to calculate the maximum velocity reach by the small rocket so this is what we are given in the question now to solve this problem have we'll be using newton's second law of motion that is sigma fx is equals to m a x and this one is our x direction this is a thrust force this is a weight force and the normal force offered by the plane so this basically is a freeboard diagram of the rocket now i can use this low here so we can write that t minus mg sine theta or you could say w sign theta is equals to m a x and from here at you have to calculate this thrust force first so thrust force is basically m dash u and m dash we have two pound per second so let me convert the unit 32 point two we're dividing with times 400 so will this get this will you to be 24 .8 pound so the thrust forces 20 24 .8 pound now this is the initial weight of the rocket but since the fuel is burned at a constant rate of 2 pound per second so the weight would be reducing as a function of time so let us first calculate the weight let us calculate the mass basically of the rocket as a function of time so it would be m not minus m dash t and m not is the initial mass so initial weight we are given to be as 125 pound so mass would be 125 divide by 322 minus m dash is 2 so again divide with 32 .2 to convert into mass 4 8 times t so we can further simplify this and we can take 1 over 32 common, 32 .2.

02:48 So here we'll get 125 minus 2t.

02:52 So this is m.

02:55 Now next the propulsion time, t.

02:59 Now the weight of the fuel is 20 pound and the rate at which the fuel is burned is 20 pounds per second...

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