The velocity $\mathrm{v}$ ms$^{-1}$ of a particle at time $t$ seconds, is given by \(\mathrm{v

Question

The velocity $\mathrm{v}$ ms$^{-1}$ of a particle at time $t$ seconds, is given by $\mathrm{v} = 2t + \cos{2t}$, for $0 \leq t \leq2$. Write down the velocity of the particle when $t = 0$. When $t = k$, the acceleration is zero. Show that $k = \frac{\pi}{4}$. Find the exact velocity when $t = \frac{\pi}{4}$. When $t < \frac{\pi}{4}$ ,$\frac{\mathrm{d}v}{\mathrm{d}t} > 0$ and when $t > \frac{\pi}{4}$,$\frac{\mathrm{d}v}{\mathrm{d}t} > 0$ Sketch a graph of $\mathrm{v}$ against $t$. Let $d$ be the distance travelled by the particle for $0 \leq t \leq 1$. Write down an expression for $d$. Represent $d$ on your sketch.

Accepted Answer

(a) $1\text{ m s}^{-1}$ (b)(i) $k=\dfrac\pi4$ (b)(ii) $\dfrac\pi2\text{ m s}^{-1}$ (c) increasing graph with stationary point of inflection $\left(\dfrac\pi4,\dfrac\pi2\right)$ (d)(i) $d=\displaystyle\int_0^1(2t+\cos2t)\,\mathrm dt=1+\dfrac12\sin2\text{ m}$ (d)(ii) the shaded area under the curve for $0\le t\le1$

Functions

Functions

Video Solution:

Solution:

Solution locked