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 1900 n. the tension in the towrope between the transport plane and the first glider is not to exceed 12000 n.

if a speed of 40 m/s is required for takeoff, what minimum length of runway is needed?

express your answer in meters to two significant figures.

Explanation:

Step1: Determine total mass

There are two gliders, each with mass \( m = 700\space kg \). Let's assume the mass of the transport plane is not given, but wait, actually, maybe we missed the plane's mass? Wait, no, maybe the problem is that the tension is 12000 N, resistance per glider is 1900 N, so total resistance for two gliders is \( 2\times1900 = 3800\space N \). The net force \( F_{net}= T - F_{resist} \), where \( T = 12000\space N \), \( F_{resist}=3800\space N \), so \( F_{net}=12000 - 3800 = 8200\space N \). Total mass \( M = 2\times700 = 1400\space kg \) (assuming only gliders? Wait, no, maybe the plane's mass is not considered? Wait, the problem says "a transport plane takes off 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 1900 N. The tension in the towrope between the transport plane and the first glider is not to exceed 12000 N." So we need to find the acceleration first. Using Newton's second law \( F_{net}=Ma \), where \( M \) is total mass of gliders? Wait, no, the tension is pulling the two gliders. So total mass of gliders is \( 2\times700 = 1400\space kg \). Net force on the gliders: \( F_{net}= T - 2\times F_{resist} \), since each glider has 1900 N resistance. So \( F_{net}=12000 - 2\times1900 = 12000 - 3800 = 8200\space N \). Then acceleration \( a=\frac{F_{net}}{M}=\frac{8200}{1400}\approx5.857\space m/s^2 \).

Step2: Use kinematic equation for distance

We know initial velocity \( u = 0\space m/s \), final velocity \( v = 40\space m/s \), acceleration \( a\approx5.857\space m/s^2 \). The kinematic equation is \( v^2 = u^2 + 2as \), where \( s \) is the distance (runway length). Since \( u = 0 \), we have \( s=\frac{v^2}{2a} \). Plugging in values: \( v = 40 \), \( a\approx5.857 \). So \( s=\frac{40^2}{2\times5.857}=\frac{1600}{11.714}\approx136.6\space m \). Wait, but maybe we missed the plane's mass? Wait, no, the tension is between plane and first glider, so the force is accelerating both gliders. So total mass is two gliders: \( 2\times700 = 1400\space kg \). Let's recalculate acceleration: \( a=\frac{12000 - 2\times1900}{1400}=\frac{12000 - 3800}{1400}=\frac{8200}{1400}\approx5.857\space m/s^2 \). Then \( s=\frac{v^2}{2a}=\frac{40^2}{2\times5.857}=\frac{1600}{11.714}\approx137\space m \), but to two significant figures, it's \( 1.4\times10^2\space m \) or 140 m? Wait, no, 136.6 is approximately 140 when rounded to two significant figures? Wait, 136.6: first significant figure 1, second 3, but wait, 136.6 to two significant figures is 1.4×10² (140) or 1.3×10²? Wait, 136.6 is closer to 140 when two significant figures? Wait, no, 136.6: the first two significant digits are 1 and 3, the next digit is 6, which is more than 5, so we round up the 3 to 4, so 1.4×10² m, which is 140 m. Wait, but let's check the calculation again.

Wait, maybe I made a mistake in the net force. The tension is 12000 N, and each glider has 1900 N resistance, so total resistance is 1900×2=3800 N. So net force on the system (two gliders) is 12000 - 3800 = 8200 N. Mass of system is 700×2=1400 kg. Acceleration a=8200/1400≈5.857 m/s². Then using v²=u²+2as, u=0, so s=v²/(2a)=1600/(2×5.857)=1600/11.714≈136.6 m. Rounding to two significant figures, that's 1.4×10² m or 140 m. Wait, but maybe the plane's mass is included? Wait, the problem says "a transport plane takes off with two gliders in tow", but the tension is between plane and first glider, so the force accelerating the gliders is the tension minus the resistance of both gliders. The plane's mass is not given, so we assume we only consider the gliders. So the calculation is as above.

Answer:

\boxed{1.4 \times 10^2} (or 140, depending on significant figures interpretation, but 136.6 rounds to 140 with two significant figures)