A Navy jet (see the figure) with a weight of 219 kN requires an airspeed of 87 m/s for liftoff. The engine develops a maximum force of 100 kN, but that is insufficient for reaching takeoff speed in the 85 m runway available on an aircraft carrier. What minimum force (assumed constant) is needed from the catapult that is used to help launch the jet? Assume that the catapult and the jet's engine each exert a constant force over the 85 m distance used for takeoff.
V^2 = Vo^2 + 2a*d
a = V^2/2d = 87^2/170 = 44.5 m/s^2.
M*g = 219,000 N.
M = 219000/g = 219000/9.8 = 22,347 kg
F = M*a = 22,347 * 44.5 =
take the answer you get for F and subtract "The engine develops a maximum force of 100 kN" to get the correct answer
To find the minimum force needed from the catapult, we can use the work-energy principle. The work done by the forces exerted by the catapult and the jet's engine should be equal to the change in kinetic energy of the jet.
The work done by a force is given by the product of force and distance:
Work = force × distance
The change in kinetic energy of the jet is given by the difference in kinetic energy between takeoff speed and initial speed:
ΔKE = 1/2 × m × (v_takeoff^2 - v_initial^2)
Given:
Weight of the jet (F_weight) = 219 kN
Maximum force developed by the engine (F_engine) = 100 kN
Takeoff speed (v_takeoff) = 87 m/s
Initial speed (v_initial) = 0 m/s
Distance for takeoff (d) = 85 m
First, let's find the weight of the jet in Newtons:
Weight = Mass × Acceleration due to gravity
F_weight = m × g
Converting the weight from kN to N:
F_weight = 219 kN × 1000 N/kN
F_weight = 219,000 N
Next, let's calculate the change in kinetic energy of the jet:
ΔKE = 1/2 × m × (v_takeoff^2 - v_initial^2)
ΔKE = 1/2 × m × (87^2 - 0^2)
ΔKE = 1/2 × m × 87^2
Since we are assuming the force exerted by the catapult and the force exerted by the engine are constant over the distance, the total work done by these forces is equal to the change in kinetic energy:
Work_total = ΔKE
The total work done is the sum of the work done by the catapult and the work done by the engine:
Work_total = Work_catapult + Work_engine
Using the formula for work (force × distance), we can rewrite the equation:
Force_catapult × distance + Force_engine × distance = ΔKE
Substituting the given values:
Force_catapult × 85m + 100 kN × 85 m = 1/2 × m × 87^2
Converting the forces from kN to N:
Force_catapult × 85m + 100,000 N × 85 m = 1/2 × m × 87^2
Simplifying the equation:
Force_catapult × 85 + 100,000 × 85 = 1/2 × m × 87^2
Now, to find the minimum force needed from the catapult, we need to rearrange the equation:
Force_catapult × 85 = 1/2 × m × 87^2 - 100,000 × 85
Simplifying further:
Force_catapult = (1/2 × m × 87^2 - 100,000 × 85) / 85
Finally, we substitute the weight of the jet (F_weight) for the mass (m) in the equation:
Force_catapult = (1/2 × F_weight/g × 87^2 - 100,000 × 85) / 85
Calculating this expression will give the minimum force needed from the catapult.
To determine the minimum force needed from the catapult, we need to find the net force acting on the Navy jet during takeoff. The net force is the difference between the thrust force generated by the engine and the friction force acting against the aircraft's motion.
1. First, let's find the friction force acting against the aircraft's motion. We can use the weight of the jet, because the force of friction is proportional to the weight. The equation for friction force is given by:
Friction Force = coefficient of friction x normal force
However, the problem does not provide the coefficient of friction. Therefore, we can assume that the frictional force is directly proportional to the weight of the aircraft.
2. Normal force is the force exerted by a surface perpendicular to the contact surface. In this case, it is equal to the weight of the aircraft since the aircraft is on a level surface (runway).
Normal Force = Weight of the Jet
3. The weight of the jet is given as 219 kN, which we can convert to Newtons.
Weight = 219 kN = 219,000 N
4. Therefore, the frictional force acting against the motion of the jet is equal to its weight, which is 219,000 N.
5. Next, let's find the net force required to accelerate the jet using Newton's second law of motion:
Net Force = mass x acceleration
We know the mass of the jet is not given, but we can calculate it using the weight of the jet and the acceleration due to gravity.
Weight = mass x acceleration due to gravity
Rearranging the equation, we find:
mass = Weight / acceleration due to gravity
The acceleration due to gravity is approximately 9.8 m/s^2.
mass = 219,000 N / 9.8 m/s^2
Solve for mass.
6. Now that we have the mass of the jet, we can find the net force required for takeoff. The only horizontal force acting on the jet is the thrust force generated by the engine.
Net Force = Thrust Force - Frictional Force
Given that the maximum thrust force the engine can generate is 100 kN, convert it to Newtons.
Maximum Thrust Force = 100 kN = 100,000 N
Substitute the values into the equation and solve for the net force.
7. Finally, to find the minimum force needed from the catapult, we subtract the net force from the thrust force.
Minimum Force from Catapult = Net Force - Thrust Force
Substitute the values into the equation and solve for the minimum force from the catapult.
Please note that the problem does not provide the values for the mass, coefficient of friction, and acceleration. Therefore, you will need to calculate or estimate those values to get an accurate answer.