Displacement and velocity from integration questions with answers

Questions

1. Question 1Displacement from velocity

   A particle moves in a straight line so that, $t$ seconds after passing through a fixed point $O$, its velocity, $v$ m/s, is given by $v = 6t^{2} + 2$. Find its displacement from $O$ when $t = 2$.

   Show the worked solution

   1. Step 1: Displacement is the integral of velocity: s = ∫ v dt (add 1 to each power, divide by the new power, add c)$s = \dfrac{6t^{3}}{3} + 2t + c$$= 2t^{3} + 2t + c$
   2. Step 2: Use the starting condition to find c$0 = 0 + c$$c = 0$
   3. Step 3: Put the given t in$s = 2(2)^{3} + 2(2)$$= 20$ m
2. Question 2Displacement from velocity

   A particle moves in a straight line so that, $t$ seconds after passing through a fixed point $O$, its velocity, $v$ m/s, is given by $v = 9t^{2} - 8t$. Find its displacement from $O$ when $t = 2$.

   Show the worked solution

   1. Step 1: Displacement is the integral of velocity: s = ∫ v dt (add 1 to each power, divide by the new power, add c)$s = \dfrac{9t^{3}}{3} - \dfrac{8t^{2}}{2} + c$$= 3t^{3} - 4t^{2} + c$
   2. Step 2: Use the starting condition to find c$0 = 0 + c$$c = 0$
   3. Step 3: Put the given t in$s = 3(2)^{3} - 4(2)^{2}$$= 8$ m
3. Question 3Velocity from acceleration

   A particle moves in a straight line so that, $t$ seconds after passing through a fixed point $O$, its acceleration, $a$ m/s², is given by $a = 6t + 1$. Its initial velocity is $10$ m/s. Find its velocity when $t = 2$.

   Show the worked solution

   1. Step 1: Velocity is the integral of acceleration: v = ∫ a dt (add 1 to each power, divide by the new power, add c)$v = \dfrac{6t^{2}}{2} + t + c$$= 3t^{2} + t + c$
   2. Step 2: Use the starting condition to find c$10 = 0 + c$$c = 10$
   3. Step 3: Put the given t in$v = 3(2)^{2} + 2 + 10$$= 24$ m/s
4. Question 4Velocity from acceleration

   A particle moves in a straight line so that, $t$ seconds after passing through a fixed point $O$, its acceleration, $a$ m/s², is given by $a = -2t + 8$. Its initial velocity is $2$ m/s. Find its velocity when $t = 2$.

   Show the worked solution

   1. Step 1: Velocity is the integral of acceleration: v = ∫ a dt (add 1 to each power, divide by the new power, add c)$v = -\dfrac{2t^{2}}{2} + 8t + c$
   2. Step 2: Use the starting condition to find c$2 = 0 + c$$c = 2$
   3. Step 3: Put the given t in$v = -(2)^{2} + 8(2) + 2$$= 14$ m/s
5. Question 5Exponential velocity

   A particle moves in a straight line so that, $t$ seconds after passing through a fixed point $O$, its velocity, $v$ m/s, is given by $v = 8e^{-2t}$. Find its exact displacement from $O$ when $t = 3$.

   Show the worked solution

   1. Step 1: Displacement: s = ∫ v dt (e to the power kt integrates to itself divided by k)$s = -4e^{-2t} + c$
   2. Step 2: Use the starting condition (it starts at O) to find c$0 = -4e^{0} + c$$c = 4$
   3. Step 3: Put the given t in$s = 4 - 4e^{-6}$
6. Question 6Exponential velocity

   A particle moves in a straight line so that, $t$ seconds after passing through a fixed point $O$, its velocity, $v$ m/s, is given by $v = 4e^{-t}$. Find its exact displacement from $O$ when $t = 3$.

   Show the worked solution

   1. Step 1: Displacement: s = ∫ v dt (e to the power kt integrates to itself divided by k)$s = -4e^{-t} + c$
   2. Step 2: Use the starting condition (it starts at O) to find c$0 = -4e^{0} + c$$c = 4$
   3. Step 3: Put the given t in$s = 4 - 4e^{-3}$