Question

A body moves with varied acceleration (a = 6t - 4), and at time (t=0), the body is moving with velocity (3 , {m/s}) and has displacement (5 , {m}) from the origin. Find, in terms of (t), the expression for the velocity and displacement of the particle.

a) (v(t) = 6t - 4), (s(t) = 3t^2 - 4t + 5)
b) (v(t) = 3t - 4), (s(t) = 6t^2 - 4t + 5)
c) (v(t) = 6t - 4), (s(t) = t^2 - 4t + 5)
d) (v(t) = 3t - 4), (s(t) = 3t^2 - 4t + 5)

Answer 1

The correct velocity function of the particle as a function of time is v(t) = 3t^2 - 4t + 3, and the correct displacement function is s(t) = t^3 - 2t^2 + 3t + 5. The provided choices all contain errors, and none match these correct expressions.

Step-by-step explanation:

The student has asked for the expression of velocity and displacement as functions of time for a particle with varied acceleration, given as a(t) = 6t - 4. At time t=0, the initial velocity is v(0) = 3 m/s and the initial displacement from the origin is s(0) = 5 m.

Step-by-step solution:

1. Integrate the acceleration function to find the velocity function. Since velocity is the integral of acceleration, we integrate a(t) = 6t - 4 with respect to t to obtain the velocity function v(t).
2. The integral of 6t is 3t^2 and the integral of -4 is -4t. Don't forget to add the constant of integration, which in this case is the initial velocity v(0) = 3.
3. So v(t) = ∫(6t - 4)dt = 3t^2 - 4t + 3.
4. Now, integrate the velocity function to find the displacement function. Since displacement is the integral of velocity, we integrate v(t) = 3t^2 - 4t + 3 with respect to t to obtain the displacement function s(t).
5. The integral of 3t^2 is t^3, the integral of -4t is -2t^2, and the integral of 3 is 3t. Again, add the constant of integration, which is the initial displacement s(0) = 5.
6. So s(t) = ∫(3t^2 - 4t + 3)dt = t^3 - 2t^2 + 3t + 5.

The correct options for velocity and displacement functions in terms of t are therefore v(t) = 3t^2 - 4t + 3 and s(t) = t^3 - 2t^2 + 3t + 5, which corresponds with none of the provided choices. The student may need to review the answer options as they contain errors.