Question 6
a) a particle moves along a horizontal line according to the equation:
x = 2t2 + 3t -5,
where x is the displacement in m) at time t (in seconds).
find the velocity and acceleration at an instant when time is 5 seconds.
(6 marks)
b) if distance s travelled by an object in time t is given by:
s = e3t
(i) calculate displacement at t = 0 seconds.
[2 marks]
(ii) prove that the acceleration is nine times distance.
(6 marks]
c) find the area enclosed by the curve y = sin 2x, the x-axis and the ordinates
x = 0 and x = t.
[6 marks

(a) The instant velocity and acceleration at a specific time are given by evaluating the first and second derivatives at

So, we have

Which implies

So, when time is 5 seconds, the object has a velocity of 23 m/s and an acceleration of 4 m/s^2 (which is indeed constant along all motion).

(b) The equation for the displacement is already given: . So, we simply have to evaluate this function at t=0 to get

So, the displacement at t=0 is 1 unit along the positive side of the line.

As for the acceleration, applying the derivation formula

we have

So, we have

and thus the acceleration is nine times the distance.

(c)

We have to evaluate the integral

The antiderivative of sin(2x) is -1/2cos(2x), becase we have

So, we have

the answer is b