Question

The functional relation between the displacement 𝑥 in meters and the time 𝑡 in seconds is given by 𝑥 = 𝐴𝑡 2 + 𝐵𝑡 + 𝐶 where 𝐴, 𝐵, and 𝐶 are constants. Find the functional relationship between the (a) velocity and the time, and (b) acceleration and the time. What are the values of the velocity and the acceleration at the instant 𝑡 = 4 𝑠𝑒𝑐𝑜𝑛𝑑𝑠

Answer 1

(a) Velocity is the derivative of the displacement with respect to time:

$$v=\dfrac{dx}{dt}=2At+B.$$

(b) Acceleration is the derivative of the velocity with respect to time:

$$a=\dfrac{dv}{dt}=2A.$$

(c)

$$v(t=4\ s)=2A\cdot4\ s=8A\ \dfrac{m}{s}.$$

(d)

$$a(t=4\ s)=2A\ \dfrac{m}{s^2}.$$