law of motion . pls help

Question

A particle moves according to the law of motion, s = f(t) = t³ −12t² + 36t, t ≥ 0 where t is measured in seconds and s in metres.

(a) Find the velocity of the particle at time t.

(b) What is the velocity after 3 seconds.

(c) When is the particle at rest?

(d) When is the particle moving forward?

(e) Find the total distance travelled by the particle during the ?rst 8 seconds.

(f) Draw a diagram to illustrate the motion of the particle.

Answer 1

So, to find velocity we take the first derivative of the equation.

 just in case you do not know how to take a derivative or a polynomial

the derivative of axⁿ  = nax^n-1  where a is a coefficient and n is the exponent.

 

s' = 3t² - 24t +36 this is your velocity in time t.

 

to solve for your velocity at t = 3 seconds we will plug 3 seconds into any t in the velocity equation.

 

s' = v this is just stating that s' is equal to your velocity

v = 3(3)² - 24(3) + 36

v = 27 - 72 + 36

v = -9 m/s

 

to find when the particle is at rest we must consider the at rest velocity which is 0. An object at rest has no velocity(only applies to Newtonian mechanics)

 

so, we will change v to 0 and solve for t

 

0 = 3t² - 24t + 36

 

to solve this we need to factor to quadratic equation.

 to solve this we first will factor out 3 from the equation.

 

0 = 3(t² - 8t + 12)

now we can factor. we need two numbers that when multiplied together give 12 but when added give -8. since the addition is negative and the multiplication is positive we know that we are dealing with two negative integers.

 

we end up with

0 = (t-2)(t-6)   the three was divided on both sides to eliminate it from the equation.

 

so, the particle is at 0 at t = 2 seconds and t = 6 seconds.

 

Now, just because the particle has 0 velocity does not mean it has no acceleration. However, I do not believe the acceleration is important for this problem.

 

To find when the particle is moving forward you must consider again when it is at 0. When it is at 0 a speed change is happening. So, to find out if velocity is positive or negative we will test a number before 2 after 2 but before 6 and after 6. 

 

So, we can start with t = 0 for the before 2 number

 

v = 3(0) - 24(0) + 36

we get v = 36 which is positive. So, the particle was moving forward before 2 seconds.

 

now we have already used 3 seconds and we got a negative velocity. So, the particle is moving backwards at t = 3 seconds.

 

So the last one to check is after 6 seconds. We will choose 10 for this one because it is easy to multiply with.

 

v = 300 - 120 + 36 = a positive number. You do not actually need to know what number as long as you know it is positive. That means the particle was moving forward after 6 seconds.

 

to find the total distance traveled we must break up the distance equation by direction of travel. So, we must take the distance traveled by the particle in pieces using the original equation.

 

s = t³ -12t² + 36t

 

first we will solve for the distance traveled in 2 seconds. 

 

s = 8 - 48 + 72 = 32 meters

 

Now we will solve for 6 seconds and figure out the distance traveled from 32 meters in the opposite direction. Then we will add the two numbers together.

 

s =216 - 432 + 192 = -24 meters

 

To find the distance travelled between 2 and 6 seconds we add the absolute value of the values together.

 

We end up with distance traveled between 2 and 6 seconds was 

 

56 meters

 

we then add this to the original 24 meters to get the distance traveled from 0 to 6 seconds of

 

88 meters

 

then we solve for the distance at 8 seconds

 

s = 512 - 768 + 288 = 32 meters

 

so, again we add the absolute values of the previous time interval with the one we just calculated for. 

 

24 + 32 = 56 meters. So, the particle traveled back up at this point.

 

We add the new 56 with the current total of 88 meters to get

 

total distance traveled  = 144 meters

 

this shows us that the particle is moving in a wave pattern from 0 to 8 seconds. So, if you were to draw a graph you would have a particle start at s = 0 and t = 0, then the particle would move much like the graph of sine instead of stopping at 1 it would stop at s = 32 then drop down to s = -24 and rise up continuously after that.