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Motion III

Motion III. Acceleration is not constant. Up until now, we have only considered motion where the acceleration is constant with respect to time. Now let us consider the general case of time dependent acceleration. Forces on a Body.

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Motion III

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  1. Motion III Acceleration is not constant

  2. Up until now, we have only considered motion where the acceleration is constant with respect to time. • Now let us consider the general case of time dependent acceleration.

  3. Forces on a Body • Consider a body of mass m subject to many time dependent force vectors. • Taking the vector sum of all the force vectors applied to the body, we get • Σ F = (t + 1)i + (t2 + 3t – 4)j + (2)k • a = [(t + 1)/m]i+ [(t2 + 3t – 4)/m]j +(2/m)k • Motion in each direction is independent of motion in the other directions, so

  4. ax= [(t + 1)/m]i • ay= [(t2 + 3t – 4)/m]j • az= (2/m)k • Remembering that ∫dv = ∫adt • We can integrate the left side from voto v • And the right side from toto t. • For the x, the y, and the z directions, but a must be kept inside the integral because it is time dependent!

  5. vx= vox + ∫[(t + 1)/m]idt • vx= vox+ (1/m)[(t2/2) + t]i • vy= voy+ ∫[( t2 + 3t – 4)/m]jdt • vy= voy+ (1/m)[(t3/3) + (3t2/2) – 4t]j • vz= voz + ∫[(-4)/m]kdt • vz= voz - (4t)/m]k

  6. The three dimensional velocity vector equation is now: • v = vx + vy + vz • v = vox+ (1/m)[(t2/2) + t]i+ voy+ (1/m)[(t3/3) + (3t2/2) – 4t]j + voz- (4t)/m]k

  7. To obtain the position vectors x, y, z, one must integrate again • ∫dx = ∫vxdt • x = xo+ ∫{vox+ (1/m)[(t2/2) + t]i}dt • x = xo+ voxt + (1/m)[(3t3/6) + (t2/2]i • ∫dy = ∫vydt • y = yo+ ∫{voy+ (1/m) )[(t3/3) + (3t2/2) – 4t]j}dt • y = yo+ voyt + (1/m) )[(t4/12) + (t3/2) – 2t2

  8. ∫dz = ∫vzdt • z = zo+ ∫{voz+ (1/m)[(-4 t]k}dt • z = zo+ vozt + (1/m)[(-2 t2]k

  9. The final position vector r is a function of time and would be the vector sum • r = x + y + z • r = xo+ voxt + (1/m)[(3t3/6) + (t2/2]i+ yo+ voyt + (1/m) )[(t4/12) + (t3/2) – 2t2]j + zo+ vozt + (1/m)[(-2 t2]k

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