Worked out Examples for Chapter 3
Worked Example A
A piano has been pushed to the top of the ramp at the back of a moving van. The
workers think it is safe, but as they walk away, it begins to roll down the ramp. The back
of the truck is 1.0 m above the ground and the ramp is inclined at 20o.
a. What is the acceleration of the piano moving down the ramp?
b. How much time do the workers have to get to the piano before it reaches the bottom
of the ramp?
Solution
Make a sketch with tilted axes with the x-axis parallel to the ramp and the angle of
inclination labeled. We must also make a bold assumption that the piano rolls down as if
it were an object sliding down with no friction.
a. The acceleration along the incline is
aincline  g sin   (9.8 m/s2 ) sin 20  3.4 m/s2
b. To determine the time it takes the piano to reach the bottom, the data table implies
that we need to determine x (the distance traveled by the piano) or the final speed of the
piano just before it reaches the bottom.
However, since the incline is described in the problem, it is easiest to
determine the distance traveled by the piano from the right triangle setup
by the incline. So the length of the ramp (or distance traveled by the piano)
is given by
h
1.0 m
x

 2.9 m
sin  sin 20o
Substituting in the distance x back into the data table, the kinematics table leads us to
the equation
2x
2  2.9m
x  v 0 t  21 aincline t 2 
t 

 1.3 s  t
aincline
3.4 m/s2
x
a
2.9 m 3.4 m/s
2
v1
vo
t
―
0 m/s
?
Worked Example B
A rifle is aimed horizontally at a target 50 m away. The bullet hits the target 2.0 cm below
the aim point.
a. What was the bullet's flight time?
b. What was the bullet's speed as it left the barrel?
Solution
Step 1: Make a drawing, define directions, and interpret the question.
Step 2,3: Find the velocity components and setup Data Tables.
o
We know v0 = ? at 0 . The components are
v 0x  v 0cosθ  v 0 ; v 0y  v 0 sinθ  0
I immediately setup data tables:
∆x
50 m
vx = vx0
?
t
−
∆y
vy
vy0
t
0-0.02 m
−
0 m/s
?
3-1
So the data table indicates that the bullet's flight time is obtained from the y-data table;
the kinematics table gives
y  v0 y t  21 gt2  t 
2y
2(0  0.02 m)

 0.064s  t
g
9.81m/s2
b. Subsituting this into the x-table, the kinematics table leads to the equation
x
50 m
x  v0x t  v 0x 

 782 m/s  v 0
t
0.064s
Worked Example C
.
a. ?
b. ?
Solution
3-2