Question

1. The force on a particle of mass 3.0 kg varies with position according to F(x) = -2.0x² (x in meters, F(x) in newtons). The particle's velocity at x = 2.0 m is 8.0 m/s. Calculate the mechanical energy of the particle using (a) the origin as the reference point and (b) x = 5.0 m as the reference point. (c) Find the particle's velocity at x = 1.0 m. Do this part of the problem for each reference point.
2. (a) How high a hill can a car coast up (engines disengaged) if work done by friction is negligible and its initial speed is 210 km/h? (b) If, in actuality, a 750-kg car with an initial speed of 210 km/h is observed to coast up a hill to a height 25.0 m above its starting point, how much thermal energy was generated by friction? (c) What is the average force of friction if the hill has a slope of 2.5° above the horizontal?

 

Answer 1

Given:

$m=3.0\:\rm kg$

$F(x)=-2.0x^2$

$v=8.0\:\rm m/s$

The mecanical energy

$$E(x)=\frac{mv^2}{2}+V(x)$$ $$V(x)=-\frac{dF}{dx}=4.0x$$

(a)

$$E(0)=E(2)=\frac{3.0*8.0^2}{2}+4.0*2.0=104\:\rm J$$

(b)

$$E(5)=E(0)=E(2)=104\:\rm J$$

(c)

$$E(1)=\frac{mv^2}{2}+V(1)=104$$

$$\frac{3.0*v^2}{2}+4.0*1.0=104\:\rm J$$

$$v=8.2\:\rm m/s$$