# Calculus Based Physic I

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**r** = b t^3 **i** + c t **j**

==> **v **= d**r**/dt = 3 b t^2 **i** + c **j**

==> **a** = d**v**/dt = 6 b t **i** + 0 **j**

==> **F** = m **a** = 6 m b t **i** = (6*20*2.1*5.5) **i** = 1386 **i** = 1400 **i**

You need to use the algebraic form of Newton's second law to evaluate the force that acts on the `20Kg` object, under the given conditions, such that:

`barF = m*bar a`

Since the problem provides the vector that indicates the position of the object, you may evaluate the acceleration vector, such that:

`a(t) = r''(t) = x''(t)bar i + y''(t) bar j`

`r(t) = bt^3bar i + ct bar j`

You need to differentiate the position vector with respect to t to evaluate the velocity vector, such that:

`v(t) = r'(t) = 3bt^2 bar i + c bar j`

You need to differentiate the velocity vector with respect to t to evaluate the accleration vector, such that:

`r''(t) = 6bt*bar i + 0`

`a(t) = r''(t) = 6bt*bar i`

You need to evaluate the force at the moment `t = 5.5 s` , hence, replacing `b = 2.1m*s^(-3)` and `5.5 s` for` t` , yields:

`a(t) = 6*2.1m*s^(-3)*5.5 s*bar i`

`a(t) = 69.3m*s^(-2)`

You need to evaluate the force, at `t = 5.5 s` , such that:

`F = 20*69.3 bar i = 1386 bar i N`

**Hence, evaluating the force, under the given conditions, yields `F = 1386 bar i N` .**

**Sources:**

F=100N

a=3m/sec^2

(i) m=?

By Newton's second law

F=m.a

100=m x 3

m=100/3 kg

(ii) Block is rest position

u=0

By second equation of linear motion

`s=ut+(1/2)at^2`

`s=0+(1/2)xx3xx10^2`

`s=(3xx100)/2=150m`

(iii)

By equation of motion

`v^2=u^2+2as`

`v^2=0+2xx3xx150`

`v^2=900`

`v=sqrt(900)`

`v=30 m//sec`

One equation for force, is F=ma.

Since we have an equation for position, we can take two derivatives to get the equation for acceleration:

Original equation:

`vecr= bt^3hati + cthatj`

First derivative (velocity):

`vecr ' = 3bt^2hati + chatj`

Second derivative (acceleration):

`vecr '' = 6bthati`

To find acceleration plug in the values for b and t:

`b=2.1m/s^3`

`t=5.5s`

`vecr '' = 6(2.1m/s^3)(5.5s)=69.3m/s^2`

Accel = 69.3 m/s^2

Mass = 20 kg

Force = ma = 69.3 m/s^2 * 20 kg = **1,386 N**

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Question asked for the force (a vector), not the magnitude of the force (a scalar). So, the answer should be a vector (1400**i** N), not a scalar (1400 N). Further, the computed value of the ith component should be reported as 1400, not 1386.