At the end of most landing runways in airports, an extension of the runway is constructed using a special substance called formcrete. Formcrete can support the weight of cars, but crumbles under the weight of airplanes to slow them down if they run off the end of a runway.
Part A
If a plane of mass 2.10×105kg is to stop from a speed of 30.0m/s on a 120m long stretch of formcrete, what is the magnitude of the average force exerted on the plane by the formcrete?
a=v²/2s
F=ma=m v²/2s =2.1•10⁵•30²/2•120 = 787500 N
Well, it sounds like the formcrete is a pretty useful substance, but it's not exactly the strongest material out there. So, let's calculate the magnitude of the average force exerted on the plane by the formcrete.
To find the average force, we can use Newton's second law of motion, which states that force equals mass times acceleration (F = ma). In this case, the acceleration is the change in velocity (since the plane is coming to a stop), divided by the time it takes to stop.
First, we need to find the change in velocity. The initial velocity (Vi) is 30.0 m/s, and the final velocity (Vf) is 0 m/s, since the plane is stopping. The change in velocity, ΔV, is therefore -30.0 m/s.
Next, we need to find the time it takes for the plane to come to a stop. We can use the formula ΔV = at, where a is the acceleration and t is the time. Rearranging the formula to solve for t, we get t = ΔV / a.
Now, we know the distance the plane travels on the formcrete (120 m) and we want to find the acceleration. We can use the equation Vf^2 = Vi^2 + 2ad, where a is the acceleration and d is the distance. Rearranging the equation to solve for a, we get a = (Vf^2 - Vi^2) / (2d).
Plugging in the values, we get a = (0^2 - 30.0^2) / (2 * 120). Simplifying, we find a ≈ -6.25 m/s^2. The negative sign indicates that the acceleration is in the opposite direction of the initial velocity.
Now we can calculate the time it takes for the plane to stop: t = (-30.0 m/s) / (-6.25 m/s^2) ≈ 4.8 s.
Finally, we can calculate the average force using F = ma. With a mass of 2.10 × 10^5 kg and an acceleration of -6.25 m/s^2, we find F ≈ (2.10 × 10^5 kg) * (-6.25 m/s^2) ≈ -1.31 × 10^6 N.
So, the magnitude of the average force exerted on the plane by the formcrete is approximately 1.31 × 10^6 N. Let's hope the formcrete holds up!
To find the magnitude of the average force exerted on the plane by the formcrete, we can use Newton's second law of motion, which states that the force exerted on an object is equal to its mass multiplied by its acceleration.
First, we need to find the acceleration of the plane. We can use the equation of motion:
v^2 = u^2 + 2as
Where:
v = final velocity (0 m/s, since the plane stops)
u = initial velocity (30.0 m/s)
a = acceleration
s = displacement (120 m)
Rearranging the equation, we have:
a = (v^2 - u^2) / (2s)
Substituting the given values, we can calculate the acceleration:
a = (0^2 - (30.0)^2) / (2 * 120)
a = -900 / 240
a = -3.75 m/s^2
Since the plane is slowing down, the acceleration is negative.
Now we can calculate the force using Newton's second law:
F = m * a
F = (2.10×105 kg) * (-3.75 m/s^2)
F = -7.88×105 N
The magnitude of the average force exerted on the plane by the formcrete is 7.88×105 N.
To find the magnitude of the average force exerted on the plane by the formcrete, we can use the concept of impulse. Impulse is defined as the change in momentum of an object and is equal to the average force applied to the object multiplied by the time over which the force acts.
In this case, the impulse experienced by the plane as it comes to a stop can be calculated using the equation:
Impulse = Change in Momentum
The change in momentum of the plane can be calculated using the equation:
Change in Momentum = Final Momentum - Initial Momentum
Since the plane comes to a stop, its final momentum is zero. The initial momentum can be calculated using the equation:
Momentum = Mass x Velocity
Therefore, the initial momentum of the plane is:
Initial Momentum = Mass x Initial Velocity
Now, let's plug in the values given in the problem:
Mass = 2.10×10^5 kg
Initial Velocity = 30.0 m/s
Initial Momentum = (2.10×10^5 kg) x (30.0 m/s)
Next, let's calculate the impulse:
Impulse = Change in Momentum = Final Momentum - Initial Momentum
Since the final momentum is zero:
Impulse = -Initial Momentum
Now we can calculate the average force exerted on the plane by the formcrete. The average force is given by:
Average Force = Impulse / Time
In this case, the time is not given explicitly. However, since we know the distance and the initial velocity, we can find the time using the equation:
Time = Distance / Initial Velocity
Now we can calculate the time:
Time = 120 m / 30.0 m/s
Finally, we can substitute the values into the equation to find the magnitude of the average force exerted on the plane by the formcrete.