Question

The position of an object is given by x = A t + B t 2, where x is in metres and t is in seconds. 

1. What are the units of A and B? [ANS A = m/s, B = m/s2] [Homogeneity analysis] 
2. What is the acceleration as a function of time? [ANS = 2B m/s2] [DIFFERENTIATION twice] 
3. What are the velocity and acceleration at t = 5.0 s? [ANS v = (A + 10B) m/s, a = 2B m/s2] 
4. What is the velocity as a function of time if x = A t + B t -3 ? [ANS = A – 3B t -4] 

 

Answer 1

1) The total units of x must be meters, so, any element (At and Bt²) must output meter. Therefore, since we have time and time squared, the units of A and B are m/s and m/s².

2) $x'(t)=A+2Bt^{2-1}=v(t),a=x''(t)=v'(t)=2B.$

3) Velocity is

$$v(5)=A+2Bt=A+10B,\\ a(5)=2B.$$

4) Velocity is

$$v(t)=A+B·(-3)·t^{-3-1}=A-3Bt^{-4}.$$