A 12 000-kg airplane launched by a catapult from an aircraft carrier is accelerated from 0 to

200 km/h in 3 s. (a) How many times the acceleration of gravity is the airplane’s acceleration? (b)
What is the average force that the catapult exerts on the airplane?

To solve this problem, we need to use the equations of motion and the concept of acceleration.

(a) To find how many times the airplane's acceleration is compared to gravity, we need to calculate the airplane's acceleration and compare it to the acceleration due to gravity.

To find the airplane's acceleration, we can use the formula:
acceleration = (final velocity - initial velocity) / time

Given:
Initial velocity (u) = 0 km/h (since it starts from rest)
Final velocity (v) = 200 km/h
Time (t) = 3 s

We first need to convert the velocities to meters per second (m/s):
1 km/h = 1000 m/3600 s = 5/18 m/s

So, the final velocity (v) = (200 km/h) * (5/18 m/s) = 500/9 m/s

Now, let's calculate the acceleration:
acceleration = (final velocity - initial velocity) / time
acceleration = (500/9 m/s - 0 m/s) / 3 s
acceleration = (500/9 m/s) / 3 s
acceleration = 500/27 m/s²

Now, to find how many times the airplane's acceleration is compared to gravity, we can compare it to the acceleration due to gravity (g).

Acceleration due to gravity (g) is approximately 9.8 m/s².

So, the ratio between the two accelerations is:
(500/27 m/s²) / (9.8 m/s²) = (500/27) / (9.8)
≈ 19.84 times

Therefore, the airplane's acceleration is approximately 19.84 times the acceleration due to gravity.

(b) To find the average force that the catapult exerts on the airplane, we can use Newton's second law of motion, which states that force (F) equals mass (m) multiplied by acceleration (a).

Given:
Mass (m) = 12,000 kg
Acceleration (a) = 500/27 m/s² (from part (a))

So, the average force (F) exerted by the catapult on the airplane can be calculated as:
F = m * a
F = 12,000 kg * (500/27 m/s²)
F ≈ 222,222.22 N

Therefore, the average force that the catapult exerts on the airplane is approximately 222,222.22 Newtons.