Q18DQ

If you know the initial position and initial velocity of a vehicle

and have a record of the acceleration at each instant, can you

compute the vehicle’s position after a certain time? If so, explain

how this might be done.

Verified

Answer

Yes the position after a certain time can be determined.

The differential equation for acceleration is $\frac{d^{2} x}{d t^{2}} = a (t)$

The initial position and initial velocity are the 2 boundary condition that are used to solve for position x(t) .

At t = 0

The position is $x (t = 0) = 50 c m$

The velocity is $x (t = 0) = 50 cmv (t) = \frac{dx}{dt} = 3 - 0.12 tcm / s$

At t = 0, the initial velocity is $v (t = 0) = 3 c m / s$

The acceleration is $a = \frac{d v}{d t} = - 0.12 c m / s^{2} a = \frac{d v}{d t} = - 0.12 c m / s^{2}$

The initial acceleration is $a (t = 0) = - 0.12 cm / s^{2} a (t = 0) = - 0.12 cm / s^{2}$

The velocity is zero when $v (t) = 3 - 0.12 t = 0 \Rightarrow t = \frac{3}{0.12} = 25 s$

$v (t) = 3 - 0.12 t = 0 \Rightarrow t = \frac{3}{0.12} = 25 s$

Hence the velocity is zero at 25 seconds.

1Step 1: Identify given quantities

Extract initial velocity, acceleration, time, displacement.

2Step 2: Choose kinematic equation

Use v=v0+at, x=v0t+0.5at^2, or v^2=v0^2+2a*dx.

3Step 3: Substitute and solve

Plug in values and solve.

4Step 4: State the answer

The answer is:

Yes the position after a certain time can be determined.

The differential equation for acceleration is $\frac{d^{2} x}{d t^{2}}$