Acceleration problems with solutions

13. A body fall from height H.if t₁ is time taken for covering first half height and
t₂ be time taken for second half.Which of these relation is true for t₁ and t₂
a. t₁ > t₂
b. t₁ < t₂
c. t₁=t₂
d. Depends on the mass of the body

Solution(13):
. Let H be the height
then
First Half
H/2=(1/2)gt₁² ----(1)
or 
(1/2)gt₁²=H/2
Also v=gt₁

Second Half
H/2=vt₂+(1/2)gt₂²
or
(H/2)=gt₁t₂+(1/2)gt₂² 
or 
(1/2)gt₂² =(H/2)-gt₁t₂ ---(2)

From 1 and 2
(1/2)gt₂²=(1/2)gt₁² -gt₁t₂ 
t₂²+2t₁t₂ -t₁²=0

or t₂=[-2t₁+√(4t₁t₂ +4t₁t₂ )]/2
or t₂=[-2t₁+2t₁√8]/2 
t₂=.44t₁

so t₁ > t₂

Hence a is correct

14.A train running at 30m/s is slowed uniformly to a stop in 44 sec.Find the stopping distance?
a. 612 m
b. 662 m
c. 630 m
d. 605 m

Solution(14):
. Here 
v₀=30m/s v_f=0 at t=44 sec
v_f=v₀+at or 0=30+a(44) or a=-.68m/s²

Now x=v₀t+(1/2)at²
or x=662 m

Hence b is correct

15.A nut comes loose from a bolt on the bottom of an elevator as the elavator is moving up the shaft at 3m/s.The nut strikes the bottom of the shaft in 2 sec.How far from the bottom of the shaft was the elevator when nut falls off?
a. 13.6 m
b. 10 m
c. 12.6 m
d. none of these

Solution(15):
. Here the nut intially has the velocity of the elevator ,so choosing upward as positive ,v₀=3 m/s
also a=-g=-9.8 m/s²

The time taken to hit the bottom is 2 s so
y=v₀t+(1/2)at²
=-13.6 m

Hence the bottom of the shaft was 13.6 m below elevator when nut fall off
Hence (a) is correct 
16. Let v and a denote the velocity and acceleration respectivly of the particle in one dimension motion.
a.the speed of the particle decreases when v.a <0
b. the speed of the particle decreases when v.a >0
c.the speed of the particle increases when v.a=0
d. the speed of the particle decreases when |v|<|a|

Solution(16):
. when v.a <0 
That mean they are in opposite direction
So when the particle is moving towards origin,acceleration is acting outwards so it is decreasing the speed
When the particle is moving outwards,acceleration is acting inwards hence it is decreasing the speed
So this option is correct

When v.a > 0
That means v and a are in same direction
When the particle is moving towards origin, acceleration is also acting inwards so increasing the speed
When the particle is moving outwards,acceleration is also acting outwards hence increasing the speed
So this option is not correct

when v.a=0
Then possible cases are
Acceleration is zero
Velocity is zero 
Both are zero
So speed does not increases in all cases.So this option is not correct

When 
|v|<|a|
Possible cases
Velocity is positive ,accleration is positive ----Speed increase
Velocity is negative ,acceleration is negative----Speed increase
Velocity is positive,acceleration is negative
Velocity is negative ,acceleration is positive---Speed decreases
So this is not correct option

So correct answer is a

More examples
Question 1 A Stone is thrown Up, what is the velocity, acceleration at the top and what direction?
Solution
At the peak,velocity of the stone is zero but accleration is not zero, acceleration is equal to acceleration due to gravity and it is acting downwards
Question 2What does the slope of the velocity-time graph shows?
Solution
Acceleration
Question 3
What does the area under the velocity time graph represent
Solution
Displacement
Question 4
Give example of the motion where x>0, v < 0 and a > 0?
Solution
x= p +qe^-t Where p and q > 0
So x > 0 at any instant
Velocity(dx/dt) = d(p +qe^-t)/dt= -qe^-t ( v < 0)
Acceleration(dv/dt) = d(-qe^-t)/dt= qe^-t ( a > 0)