1D Kinematics Solved Examples

Question 17.The displacement of a particle moving in straight line depends on time t as
x=at³+bt²+ct+d

which of the follwing is true
a. Intial acceleration depends on b only
b Intial velocity depends on c only
c. Intial displacement is d
d. Ratio of intial velocity /intial acceleration depends on a and c

Solution(17):
. The displacement of a particle moving in straight line depends on time t as
x=at³+bt²+ct+d

Velocity (dx/dt)=3at²+2bt+c
Acceleration (d²x/dt²)=6at+2b

So Intial displacement(t=0) =d
Intial velolcity(t=0) =c
Intial acceleration (t=0) =2b

Question 18.A particle located at x=0 at time t=0 starts moving along the positive x-direction with a velocity v that varies as
v=√x
The displacement of the particles varies with time as 
which of the follwing is true
a. t²
b t³
c. t⁴
d. t^1/2

Solution(18):
. Given v=√x
or
dx/dt=√x
dx/√x=dt 
Integrating both sides between the limit (0,x) and (0,t)
x=t²/4
Hence (a) is correct
Question 19.A Train is moving along a straight section of the track with a velocity of 180km/h. The braking deceleration is 2m/s²s
At what distance from a train station should the train driver aply the brake so that train stops at the station 
a. 800m
b 625m
c. 700m
d. none of these
Solution(19):
. we are given 
v₀=180km/h=50m/s
a=-2m/s²
Now v²=v₀²+2as 
0=(50)²-2*2*x or x=625m hence (b) is correct Question 20.from the previous question,how long will it take to bring the train to the halt 
a. 25s
b 20s
c. 15s
d. none of these
Solution(20):
. Now v=v₀+at
0=50-2t
or t=25 sec
Hence (a) is correct

More Practice Questions 
Question 1 A car brakes from a speed of 108 km/h to 72 km/h duing a displacement of 100m. What is its acceleration?
Solution This can be easily found using Third Kinematics equations
v²=u²+2as
a=-2.5 m/s²
Question 2 A Ball is dropped from a building of the height 90 m . Simultanously another ball is thrown up with a speed of 40 m/s. Calculate the Relative speed of the balls as a function of time?
Solution
Both the balls are falling freely under gravity .Therefore no acceleration of one with respect to other .So relative speed of the balls remains constant equal to 40m/s
Question 3
The Relation between t and distance x is given by 
t=ax² +bx where a and b are constant. 
Expreess instaneous acceleration in terms of instaneous velocity
Solution
acceleration =-2av³
Question 4 A car travels 100 km in 2 hours and 50 km in 1 hour in the same direction. 
a) What is the Car's average speed for the whole journey? 
b) What is the Car's average velocity for the whole journey?
Solution
50km/h,50km/h
Question 5
Two balls of different masses (one lighter and other heavier) are thrown vertically upwards; with the same speed. Which one will pass through the point of projection in their downward direction with the greater speed?
Solution
In case of motion under gravity, the speed with which a body returns back is always equal to the speed with which it is thrown up. Since expression for final speed does not involve mass, both the balls will acquire the same speed.