Deriving Equations for Laws of Motion

Aug 2015
10
0
USA
Guys I am trying to teach myself calculus, and I have stumbled across these manuals my dad had and there are a few problem series that I need help with.

Any suggestions or direction to help outside the forum is appreciated.

1) Suppose a body "A" does not begin at rest, but instead starts with an initial velocity of 10 meters per second downward at the point B at time t=0. Again let s be the distance AB. Use calculus to show that

(a) if the acceleration is equal to a constant g=9.8 meters per second per second, then

s= 10t + 4.9t^2

(b) if s= 10t + 4.9 t^2, then the acceleration is equal to 9.8meters per second per second and the velocity

v=10 + 9.8t

2) Suppose that the body begins at a position 1 meter above the point B with an upward velocity of 15 meters per second. Find an equation for s in terms of t.

3) Suppose that the body begins at rest at the point B , but now suppose that the force is not constant. Instead, suppose that the body is acted on by a force causing a downward acceleration

a=2t + 1

Find an equation for s in terms of t

4) Suppose that acceleration is given by the equation

a= sin t

(a) If the body begins at point B at rest at ti t=0, find an equation for s in terms of t

(b) If the body begins at point B with a downward velocity of 5 meters per second, find an equation for s in terms of t.
Hey can you help me deriving these equations?

Suppose a body "A" does not begin at rest, but instead starts with an initial velocity of 10 meters per second downward at the point B at time t=0. Again let s be the distance AB. Use calculus to show that

(a) if the acceleration is equal to a constant g=9.8 meters per second per second, then

s= 10t + 4.9t^2

(b) if s= 10t + 4.9 t^2, then the acceleration is equal to 9.8meters per second per second and the velocity

v=10 + 9.8t

2) Suppose that the body begins at a position 1 meter above the point B with an upward velocity of 15 meters per second. Find an equation for s in terms of t.

3) Suppose that the body begins at rest at the point B , but now suppose that the force is not constant. Instead, suppose that the body is acted on by a force causing a downward acceleration

a=2t + 1

Find an equation for s in terms of t

4) Suppose that acceleration is given by the equation

a= sin t

(a) If the body begins at point B at rest at time t=0, find an equation for s in terms of t.

(b) If the body begins at point B with a downward velocity of 5 meters per second, find an equation for s in terms of t.
 
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Aug 2015
10
0
USA
I think I understand but I have such low confidence. This was what I did for the first one:

I think I am taking integral from 0 to v. I have no idea how to represent that on here:

v= gdt
∫vdt= ∫([10,v])g(t)dt
s= ∫g (v) dt - ∫g (10) dt
= g∫vdt -g∫10dt
= s - 10t
= 1/2gt^2 +10t (Am I allowed to make this substitution?)
= 1/2 (9.80) t^2 +10t
= 4.9t^2 + 10t


s = 10t +4.9t^2
d/ds = d/dt(10t + 4.9t^2)
v = 10d/dt(t) + 4.9d/dt(t^2)
= 10 + 4.9(2t)
= 10 + 9.8t

Are these correct? If so, can you help me with the others?
 
Last edited:

Country Boy

Math Team
Jan 2015
3,261
899
Alabama
I think I understand but I have such low confidence. This was what I did for the first one:

I think I am taking integral from 0 to v. I have no idea how to represent that on here:

v= gdt
Since acceleration is the derivative of velocity with respect to time, dv/dt= a.
The acceleration due to gravity is -g (negative because you are measuring height upward and acceleration is downward) so dv/dt= - and so
dv= -g dt.

∫vdt= ∫([10,v])g(t)dt
Almost. \(\displaystyle \int dv= \int -g dt\). Also your limits of integration on the right make no sense. "10" and "v" are values of speed, not time. They cannot be limits of integration for integration with respect to time. They could be limits of integration for the integral on the left. You would do better to do this as an indefinite integral, v= -gt+ C, and use the initial condition, that v(0)= -10 (again, negative because the initial speed is downward) to get C= -10: v= -gt- 10

s= ∫g (v) dt - ∫g (10) dt
Writing "g(v)" makes it look like g is a function of t - just leave it as gv
But v= ds/dt so ds= vdt= (-gt- 10)dt. Integrating both sides \(\displaystyle v= -(g/2)t^2- 10t+ C\)

= g∫vdt -g∫10dt
= s - 10t
= 1/2gt^2 +10t (Am I allowed to make this substitution?)
No, because s is not equal to (1/2)gt^2! It is s you are trying to find.

= 1/2 (9.80) t^2 +10t
= 4.9t^2 + 10t


s = 10t +4.9t^2
d/ds = d/dt(10t + 4.9t^2)
v = 10d/dt(t) + 4.9d/dt(t^2)
= 10 + 4.9(2t)
= 10 + 9.8t

Are these correct? If so, can you help me with the others?
It is not clear what "C" in \(\displaystyle s= -(1/2)gt^2- 10t+ C\) should be because it is not clear what the initial height is. You say "Suppose a body "A" does not begin at rest, but instead starts with an initial velocity of 10 meters per second downward at the point B at time t=0." If you are taking point B to be s= 0 then C will be 0 and the distance from B to A will be \(\displaystyle s= -(1/2)gt^2- 10t\). That is negative because A is below B.
 
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Sep 2015
17
4
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Order yourself a copy of "Calculus Made Easy" by Silvanus P Thompson. Quickest way to learn basic calculus.