Car 1 had mass 1000 kg and speed 6 m/s. It collided with Car 2, which had been a

Question

Car 1 had mass 1000 kg and speed 6 m/s. It collided with Car 2, which had been at rest, and they stuck together. The cars then moved off together with speed 2 m/s. What was the mass of Car 2? The force acting on a particle is F_x = 1/3x^2 + x. Assume SI units are being used. Find the work done by this force as the particle moves from x_i= 1 m to x = 3 m . A wheel has initial angular velocity w_0 = 30 rad/s and then undergoes an angular acceleration a - 0.6t 3t for 12 s. Find the angular velocity at the end of this time. (Assume SI units are used.) For the following potential energy function, find the force components (as functions of x and y). U(x,y) = 0.5x^3y - 4x^2y^4 + 1.8y

Explanation / Answer

1) m1 =1000 kg , u1 =6 m/s u2 =0, v = 2m/s

From conservation of momentum

m1u1+m2u2 = (m1+m2)v

(1000*6) +0 = (1000+m2)2

mass of car 2 is m2 =2000 kg

2) Fx = (1/3)x^2 +x

x1 =1 m , x2 = 3m

inetegration of Fx gives work done

W = (1/3)(x^3/3) + (x^2/2)

w = (x^3/9) +(x^2/2)

W = (1/9)[ (3)^3 -(1)^3] +(1/2)[(3)^2 - (1)^2]

w =6.89 J

3) wo = 30 rad/s

alpha =0.6t^3 - 3t^2

integration of alpha gives angular veloctiy

w -wo = 0.6(t^4/4) -3(t^3)/3

at t= 12 s

w-30 = (0.15)(12)^4 - (12)^3

w =1412.4 rad/s

4) U(x,y) =0.5x^3y -4x^2y^4 +1.8 y

derivation U(x,y) gives the force

Fx =dU/dx

Fx = 1.5x^2y - 8xy^4 +1.8y

5) Fy =dU/dy

Fy = 0.5x^3 -16x^2y^3 +1.8

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