Q18E - University Physics with Modern Physics

Question

A transport plane takes off from a level landing field with two gliders in tow, one behind the other. The mass of each glider is 700 kg , and the total resistance (air drag plus friction with the runway) on each may be assumed constant and equal to 2500N . The tension in the towrope between the transport plane and the first glider is not to exceed . (a) If a speed of 40 m/s  is required for take off, what minimum length of runway is needed? (b) What is the tension in the towrope between the two gliders while they are accelerating for the take off?

Step-by-Step Solution

Answer

(a) The minimum length of runway needed is 160 m .

(b) The tension in the tow rope is 6000 N . 

Step 1: Determine the acceleration of the system

Given Data:

The mass of each glider is $m=700 \mathrm{kg}$ 

The total air resistance on each glider is $f=2500 \mathrm{N}$  

The tension in the rope between transport plane and glider is $T_1=12000 \mathrm{N}$  

The speed for takeoff of plane is: $v=40 \mathrm{m}/\mathrm{s}$ 

Step 2: Air Resistance:

The opposing force offered by air on the moving object in air is called resistance force. This opposing force varies with the speed of the object. If the vector sum of the forces on the body is not zero then the body will accelerates and it is said to be in non-inertial frame of reference.

Step 3: Determine the acceleration of the system

(a)

The acceleration of the plane is calculated as:

$T_1-2f=2ma$ 

Substitute  12000N for $T_1, 2500\mathrm{N}$ for $f$ and 700 kg  for $m$ in the above equation.

$$\begin{gathered} 12000\mathrm{N}-2\left(2500 \mathrm{N}\right)=2\left(700 \mathrm{kg}\right)\mathrm{a} \\ \mathrm{a}=5 \mathrm{m}/{\mathrm{s}}^2 \end{gathered}$$ 

The minimum length of runway needed is calculated as:

$v^2=u^2+2al$ 

Here, u  is the initial speed of plane and its value is zero,  a is the acceleration of the system of plane, $l$ is the minimum length of runway

Substitute $40\mathrm{m}/\mathrm{s}$ for v ,  0m/s for u and $5\mathrm{m}/{\mathrm{s}}^2$ for a  in the above equation.

$$\begin{gathered} {\left(40 \mathrm{m}/\mathrm{s}\right)}^2={\left(0\right)}^2+2\left(5 \mathrm{m}/{\mathrm{s}}^2\right)l \\ l=160 \mathrm{m} \end{gathered}$$ 

Therefore, the minimum length of runway needed is 160 m.

Step 4: Determine the tension in the tow rope

(b)

The tension in the tow rope is calculated as:

$T=f+ma$  

Here, $T$ is the tension in tow rope.

Substitute 2500 N for $f$ , 700 kg  for  $m$ and $5 \mathrm{m}/{\mathrm{s}}^2$  for $a$  in the above equation.

$$\begin{gathered} T=2500 \mathrm{N}+\left(700 \mathrm{kg}\right)\left(5 \mathrm{m}/{\mathrm{s}}^2\right) \\ T=6000 \mathrm{N} \end{gathered}$$ 

Therefore, the tension in the tow rope is 6000 N .