The kinetic energy KE of an object of mass m moving with velocity v is defined as KE -m2. If a force fx) acts on the object, moving it along the x-axis from xi to x2, the Work-Energy Theorem states that the net weight equal to the change in kinetic energy: mv2 - mv1, where V1 is the velocity at xi and v2 is the velocity at x2. Suppose that when launching a 700-kg roller coaster car an electromagnetic propulsion system exerts a force of 5.7x2 + 1.5x newtons on the car at a distance x meters along the track. Use the Work-Energy Theorem to find the speed of the car when it has traveled 60 meters.

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:

[tex]W=\Delta K=\frac{1}{2}mv_2^2-\frac{1}{2}mv_1^2\\\\W=\frac{1}{2}m(v_2^2-v_1^2)[/tex] (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:

[tex]W=\int_0^{60} Fdx[/tex] (2)

where F is given by:

[tex]F(x)=5.7x^2+1.5x[/tex]

You replace the previous function F in the integral (2) and calculate W:

[tex]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[/tex]

Next, you solve the equation (1) for v2, and replace the values of all parameters:

[tex]W=\frac{1}{2}mv_2^2+0\\\\v_2=\sqrt{\frac{2W}{m}}=\sqrt{\frac{2(413100J)}{700kg}}=34.35\frac{m}{s}[/tex]

The final speed of roller coaster, after it has traveled 60m is 34.35m/s