Answer:
v = 34.35 m/s
Explanation:
In order to find the speed of the car, when it has traveled 60m, you take into account the Work-Energy theorem, which is given by the following formula:
(1)
W: work done on the roller coaster by the electromagnetic propulsion
m. mass of the roller coaster = 700kg
v2: final speed of the roller coaster = ?
v1: initial speed = 0m/s
you have that the force exerted on the roller depends on x, the distance traveled by the roller. To calculate the work W, you use the following integral:
(2)
where F is given by:
![F(x)=5.7x^2+1.5x](https://tex.z-dn.net/?f=F%28x%29%3D5.7x%5E2%2B1.5x)
You replace the previous function F in the integral (2) and calculate W:
![W=\int_0^{60}(5.7x^2+1.5x)dx=[\frac{5.7x^3}{3}+\frac{1.5x^2}{2}]_0^{60}\\\\W=[1.9(60)^3+0.75(60)^2-0]=413100\ J](https://tex.z-dn.net/?f=W%3D%5Cint_0%5E%7B60%7D%285.7x%5E2%2B1.5x%29dx%3D%5B%5Cfrac%7B5.7x%5E3%7D%7B3%7D%2B%5Cfrac%7B1.5x%5E2%7D%7B2%7D%5D_0%5E%7B60%7D%5C%5C%5C%5CW%3D%5B1.9%2860%29%5E3%2B0.75%2860%29%5E2-0%5D%3D413100%5C%20J)
Next, you solve the equation (1) for v2, and replace the values of all parameters:
![W=\frac{1}{2}mv_2^2+0\\\\v_2=\sqrt{\frac{2W}{m}}=\sqrt{\frac{2(413100J)}{700kg}}=34.35\frac{m}{s}](https://tex.z-dn.net/?f=W%3D%5Cfrac%7B1%7D%7B2%7Dmv_2%5E2%2B0%5C%5C%5C%5Cv_2%3D%5Csqrt%7B%5Cfrac%7B2W%7D%7Bm%7D%7D%3D%5Csqrt%7B%5Cfrac%7B2%28413100J%29%7D%7B700kg%7D%7D%3D34.35%5Cfrac%7Bm%7D%7Bs%7D)
The final speed of roller coaster, after it has traveled 60m is 34.35m/s