The acceleration of a particle at time #t# seconds is given by # a = 6t#. Given that #v=10# when #t=0# and that #x=7# when #t=0# find #x# when #t=2#?

Question

The acceleration of a particle at time #t# seconds is given by # a = 6t#.  Given that #v=10# when #t=0# and that #x=7# when #t=0# find #x# when #t=2#?

Emilia Aguilar · Accepted Answer

The position #x# at time #t# is given by;

# x = t^3 + 10t + 7 #

And so when #t=2# we have:

# x = 35#

Acceleration (#a#) is defined as the rate of change of velocity (#v#) wrt time (#t#). thus # a = (dv)/dt => (dv)/dt = 6t# This is a First Order separable Differential Equation which we can just integrate to get: # v = 3t^2 + A # Using the initial condition #v=10# when #t=0# we get: # \ \ \ 10 = 0 + A => A=10 # # :. v = 3t^2 + 10 # Velocity (#v#) is defined as the rate of change of position (#x#) wrt time (#t#). thus # v = (dx)/dt => (dx)/dt = 3t^2 + 10# Again this is a First Order separable Differential Equation which we can just integrate to get: # x = t^3 + 10t + B # And using the initial condition #x=7# when #t=0# we get: # 7 = 0 + 0 + B => B= 7# Hence the position #x# at time #t# is given by; # x = t^3 + 10t + 7 # And so when #t=2# we have: # x = 2^3 + 10*2 + 7 =35#

Paisley Johnson · Answer

undefined To find $x$ when $t = 2$, integrate the acceleration function $a = 6t$ to get the velocity function $v(t)$, then integrate the velocity function to get the displacement function $x(t)$. Given that $v = 10$ when $t = 0$, you can find the constant of integration for velocity, and since $x = 7$ when $t = 0$, you can find the constant of integration for displacement. Finally, plug in $t = 2$ into the displacement function to find $x$.

1. Integrate $a = 6t$ to find $v(t)$:
$$v(t) = \int 6t dt = 3t^2 + C_1$$

2. Use the given condition $v = 10$ when $t = 0$ to find $C_1$:
$$10 = 3(0)^2 + C_1 \Rightarrow C_1 = 10$$

3. Integrate $v(t) = 3t^2 + 10$ to find $x(t)$:
$$x(t) = \int (3t^2 + 10) dt = t^3 + 10t + C_2$$

4. Use the given condition $x = 7$ when $t = 0$ to find $C_2$:
$$7 = (0)^3 + 10(0) + C_2 \Rightarrow C_2 = 7$$

5. Now, plug in $t = 2$ into the displacement function to find $x$:
$$x(2) = (2)^3 + 10(2) + 7 = 8 + 20 + 7 = 35$$

So, $x = 35$ when $t = 2$.