2) The ancient Calmarians arc about to invade the Nishmatites. In order to do so

Question

2) The ancient Calmarians arc about to invade the Nishmatites. In order to do so. they have to 2) _ get their soldiers quickly across the river, which is 30.0 m wide. They therefore designed catapults consisting of a spring and a ramp that launches a horseless wagon across the river, as shown in the figure. The ramp has a 15.0 degree grade and a 20.0 in horizontal run Each wagon has a mass of 115 kg. including passenger. The spring has an equilibrium length of 2.00 m. and is compressed to a length of 1.00 m in order to launch the wagon. What is the minimum spring constant that the spring must have in order for the wagons to make it safely across the river? (Assume that there is no friction anywhere. Remember to consider the filet that the initial position of the wagon is nor at the base of the ramp due to the finite length of the compressed spring.) 311.11 m Figure 2 - A catapult consisting of a ramp with a 15.0 degree grade and a 20.0 in horizontal run. and a spring wide an equilibrium length of 2.00 m. is used to launch a horseless wagon across the river. I he mass of the wagon. including passenger. is 115 kg. The river is 30.0 m wide.

Explanation / Answer

Here , for the wagon to cross the river , as

the equation of trajectory is given as

y = y0 + x*tan(theta) - g*x^2/2(v*cos(theta))

0 = 1.5 + 30*tan(15) - 9.8*30^2/(2*(v*cos(15))^2)

solving for v

v = 22.3 m/s

Now . using conservation of energy

0.5 * k*x^2 = 0.5*mv^2 + mgh

0.5 * k * 1^2 = 0.5 * 115 * 22.3^2 + (20*tan(15) - 1*sin(15))*115 * 9.8

solving for k

k = 68684 N/m

the spring constant is 68684 N/m

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