Question

particles p and Q leave a fixed point A at the same time and travel in the same straight line. the velocity of P after t seconds is 6t(t-3)ms-1 and the velocity of Q after t seconds is (10-2t)ms-1.

1) sketch,on the same axes, velocity-time graph for P and Q for 0<=t<=5

2)verify that P and Q meet after 5 seconds.

3)find the greatest distance between P and Q for 0<=t<=5

Answer 1

1)

2) After 5 seconds:

$$S_1=\int v(t)dt=\int(6t^2-18t)dt=(2t^3-9t^2)|_0^5=25$$

$$S_2==\int v(t)dt=\int(10-2t)dt=(10t-t^2)|_0^5=25$$

3)

$$f(t)=v_2t-v_1t$$

$$f(t)=10t-2t^2-6t^3+18t^2$$

$$f^/(t)=-18t^2+32t+10$$

$$t \approx1$$

$$f(1)=20$$