A navy jet with a weight of 231 kN requires an airspeed of 85 m/s for liftoff. The engine develops a maximum force of 107 kN, but that is insufficient for reaching takeoff speed in the 90 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 90 m distance useed for takeoff.
I can't figure out the equation for this problem to get me started.
There is no single formula to memorize and look up to do this problem. You have to do it one step at a time, using formulas you should know.
The plane has to develop a speed of V = 85 m/s in X = 90 meters. That requires an acceleration a given by
V = sqrt (2 a X)
Solve for a.
a = V^2/(2X)
The total force needed to accelerate the plane is given by F = ma, where m = (231/kN/9.8 m/s^2) = 23570 kg.
Compute F.
There is already 107 kN of force provided by the engimes. The difference between F and that number must be provided by the catapult.
Please try to fill in the missing steps. If you need additional help, show your work.
Sure! Let's go step-by-step to solve the problem.
Step 1: Calculate the acceleration needed
We can use the formula V = sqrt(2 a X) to calculate the acceleration required to reach a speed of 85 m/s in a distance of 90 meters.
V = 85 m/s (given)
X = 90 m (given)
a = ?
Rearranging the formula, we get:
a = V^2 / (2X)
Substituting the given values, we have:
a = (85^2) / (2 * 90)
Calculating this, we find:
a ≈ 40.06 m/s^2
Step 2: Calculate the total force required
To find the total force required, we can use the formula F = ma, where m is the mass of the jet.
m = 231 kN / (9.8 m/s^2)
m ≈ 23570 kg
F = ?
a ≈ 40.06 m/s^2 (from Step 1)
m ≈ 23570 kg (calculated)
Using the formula F = ma, we substitute the values:
F = 23570 kg × 40.06 m/s^2
Calculating this, we find:
F ≈ 943641.2 N
Step 3: Calculate the force provided by the engines
Given that the engine provides a maximum force of 107 kN, we need to convert it to Newtons.
1 kN = 1000 N
So, force provided by the engine = 107 kN × 1000 N/kN
force provided by the engine = 107,000 N
Step 4: Calculate the force required from the catapult
The minimum force required from the catapult is the difference between the total force required and the force provided by the engines.
Force required from the catapult = Total force required - Force provided by the engines
Force required from the catapult = 943641.2 N - 107,000 N
Calculating this, we find:
Force required from the catapult ≈ 836641.2 N
So, the minimum force required from the catapult to help launch the jet is approximately 836,641.2 Newtons.
To solve this problem, we can break it down into the following steps:
Step 1: Calculate the acceleration (a) required to reach the takeoff speed.
We can start by using the equation: V = sqrt(2aX), where V is the takeoff speed (85 m/s) and X is the distance (90 m).
V = sqrt(2aX)
85 = sqrt(2a * 90)
To solve for a, we square both sides of the equation and isolate "a".
a = (85^2)/(2 * 90)
a = 7225/180
a ≈ 40.14 m/s^2
Step 2: Calculate the total force (F) required to accelerate the plane.
The total force (F) can be calculated using the formula F = ma, where m is the mass of the plane, and a is the acceleration.
Given that the weight of the plane is 231 kN, we can determine the mass (m) in kilograms by dividing the weight by the acceleration due to gravity (9.8 m/s^2).
m = 231 kN / 9.8 m/s^2
m = 23570 kg
Now we can calculate the total force (F).
F = ma
F = 23570 kg * 40.14 m/s^2
F ≈ 945 kN
Step 3: Calculate the minimum force required from the catapult.
The maximum force provided by the engines is given as 107 kN. To find the minimum force required from the catapult, we subtract the force provided by the engines (107 kN) from the total force required (945 kN).
Minimum force from the catapult = Total force required - Force provided by the engines
Minimum force from the catapult = 945 kN - 107 kN
Minimum force from the catapult ≈ 838 kN
Therefore, for the navy jet to achieve liftoff using the available runway on the aircraft carrier, a minimum force of approximately 838 kN is required from the catapult.