A particle travels along a straight line with its acceleration at time t secs = (3t + 2)m/s^2. Find velocity of the body when t=5.

Question

Particle has an initial positive velocity & travels 30m in the 4th second.

This is my working out so far, but I'm not sure how to go from there. Any help on how to go about this problem would be appreciated , thanks in advance!

Given a = 3t+2:

When t=4, x (displacement)=30

v= ∫ a dt

= ∫ 3t +2 dt

= 1.5t^2 + 2t + c

x (displacement) = ∫ velocity dt

             = 0.5t^3 + t^2 + ct + g (where g is another constant)

when t=4:

How do you deduce the constants c and g?

This question is from Sadler Methods Unit 3 (Year 12).

Answer 1

$a=3t+2 \Rightarrow v = \dfrac{3}{2}t^2+2t+v_0 \Rightarrow x=\dfrac{1}{2}t^3+t^2+v_0t+x_0$

$v_0$ and $x_0$ are respectively the initial velocity and position.

You want $v(5) = \dfrac{75}{2}+10+v_0=\dfrac{95}{2}+v_0$.

You know $x(4)-x_0=30 \, \mathrm{m}$ because you said that the body has travelled $30 \, \mathrm{m}$ in 4 seconds (from the inital time $t=0 \ \mathrm{s}$).

So $64/2+16+4v_0+x_0-x_0=30\Rightarrow 18 = 4v_0 \Rightarrow v_0 = \dfrac{18}{4} = \dfrac{9}{2}$.

Therefore $v(5) = \dfrac{95}{2} + \dfrac{9}{2} = \dfrac{104}{2} = 52 \, \mathrm{m.s^{-1}}$.

I hope it helps you.

Answer 2

You can consider the problem as a solving second order differential equations and inserting the initial values for the in order will give you the constants. İnitially, consider the Newton's second law for your system then you will get the g term and using the initial speed you will get the other constant. Basically, write the equation of motion for your system with step by step integration and inserting the initial values in order.