A particle, P, starts moving in a straight line with a velocity of 8 m/s at a displacement of 3 m from a fixed point O. The velocity, $v$ m/s of the particle $t$ seconds after moving is given by $v = 5\cos t + 2\sin t + c$.

1. Substitute the initial conditions into the velocity equation

At $t = 0$, $v = 8$.
$8 = 5\cos(0) + 2\sin(0) + c$

2. Evaluate the trigonometric functions and solve for $c$

$8 = 5(1) + 2(0) + c$
$8 = 5 + c$
$c = 3$

1. Differentiate the velocity function to find the acceleration function

$a = \frac{dv}{dt} = \frac{d}{dt}(5\cos t + 2\sin t + 3)$
$a = -5\sin t + 2\cos t$

2. Substitute $t = 0$ to find the initial acceleration

$a(0) = -5\sin(0) + 2\cos(0)$
$a(0) = 0 + 2(1) = 2$

1. State the velocity function using the value of $c$ found in part (a)

From part (a), $c = 3$, so the velocity function is:
$v(t) = 5\cos t + 2\sin t + 3$

2. Calculate the velocity at the start and end of the 3rd second ($t=2$ and $t=3$)

The 3rd second corresponds to the time interval $2 < t < 3$.
At $t = 2$:
$v(2) = 5\cos 2 + 2\sin 2 + 3 \approx 2.74 \text{ m/s}$
At $t = 3$:
$v(3) = 5\cos 3 + 2\sin 3 + 3 \approx -1.67 \text{ m/s}$

3. Conclude that the change in sign of velocity implies a change in direction

Since $v(2) > 0$ and $v(3) < 0$, the velocity changes from positive to negative between $t = 2$ and $t = 3$. Therefore, the particle changed its direction of motion during the 3rd second.

1. Integrate the velocity function to find the general expression for displacement

From part (a), $c = 3$, so $v = 5\cos t + 2\sin t + 3$.
Let $s$ be the displacement.
$s = \int v \, dt$
$s = \int (5\cos t + 2\sin t + 3) \, dt$
$s = 5\sin t - 2\cos t + 3t + k$

2. Use the initial condition to find the constant of integration

Given that at $t = 0$, $s = 3$:
$3 = 5\sin(0) - 2\cos(0) + 3(0) + k$
$3 = 0 - 2(1) + 0 + k$
$3 = -2 + k$
$k = 5$
So, $s = 5\sin t - 2\cos t + 3t + 5$.

3. Calculate the displacement at $t = 6$

Substitute $t = 6$ into the displacement equation (ensure calculator is in radian mode):
$s = 5\sin(6) - 2\cos(6) + 3(6) + 5$
$s = 5(-0.2794) - 2(0.9602) + 18 + 5$
$s = -1.397 - 1.920 + 23$
$s = 19.683$
$s \approx 19.7 \text{ m (to 3 s.f.)}$