5.13 KinematicsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   A particle moves in a straight line. Its velocity at time $t$ seconds is $v=3t^2-12t+9$ m s$^{-1}$, for $t\ge0$.

   (a)

   Find the acceleration of the particle when $t=4$.

   [1 mark]

   • A$9$ m s$^{-2}$
   • B$24$ m s$^{-2}$
   • C$12$ m s$^{-2}$
   • D$36$ m s$^{-2}$

   (b)

   At which times is the particle instantaneously at rest?

   [1 mark]

   • A$t=1$ and $t=3$
   • B$t=3$ only
   • C$t=1$ and $t=9$
   • D$t=0$ and $t=4$

   (c)

   Find the displacement of the particle between $t=0$ and $t=2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle moves along a straight line. When its displacement from the origin is $s$ metres, its velocity is $v=s^2-4$ m s$^{-1}$.

   (a)

   Find the acceleration of the particle when $s=3$.

   [1 mark]

   • A$6$ m s$^{-2}$
   • B$5$ m s$^{-2}$
   • C$60$ m s$^{-2}$
   • D$30$ m s$^{-2}$

   (b)

   Find the speed of the particle when $s=1$.

   [1 mark]

   • A$-3$ m s$^{-1}$
   • B$3$ m s$^{-1}$
   • C$-6$ m s$^{-1}$
   • D$5$ m s$^{-1}$

   (c)

   Find the acceleration of the particle when $s=-1$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A robot moves along a straight rail. Its velocity at time $t$ seconds is $v=4\cos(0.5t)$ m s$^{-1}$ for $0\le t\le2\pi$, where the angle is in radians. A GDC may be used.

   (a)

   Find the first time at which the robot is at rest, and its acceleration at that time.

   [3 marks]

   (b)

   (i) Find the displacement of the robot from $t=0$ to $t=2\pi$.
   (ii) Find the total distance travelled by the robot from $t=0$ to $t=2\pi$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle moves along the $x$-axis. At time $t$ seconds its velocity is $\dot{x}=t^2-7t+10$ m s$^{-1}$ for $0\le t\le6$. Initially $x=3$ m.

   (a)

   (i) Find the times at which the particle is instantaneously at rest.
   (ii) Find $\ddot{x}$ when $t=3$, and hence state, with a reason, whether the particle is speeding up or slowing down when $t=3$.
   (iii) Find the speed of the particle when $t=3$.

   [6 marks]

   (b)

   (i) Find the displacement of the particle from $t=0$ to $t=6$.
   (ii) Find the total distance travelled by the particle from $t=0$ to $t=6$.
   (iii) Find the value of $x$ when $t=6$.

   [6 marks]

   Total for question 4: 12 marks

End of questions