Motion in Two Dimensions
Projectile Motion
A projectile is an object moving in two dimensions under the influence of Earth's gravity; its path is a parabola.
Motion in Two Dimensions
ay = g
ax = 0
Motion in Two Dimensions
Motion in Two Dimensions
Ignoring air resistance, the horizontal component of a projectile's acceleration
(A) is zero.
(B) remains a non-zero constant.
(C) continuously increases.
(D) continuously decreases.
If an object is launched at an initial angle of θ0 with the horizontal, the analysis is similar except that the initial velocity has a vertical component.
xv
yv
ovxv
yv1v
xv2v
xv
yv3v
xv
yv
fv
31 vv
fo vv
Motion in Two Dimensions
Motion in Two Dimensions
Ignoring air resistance, the horizontal component of a projectile's velocity
(A) is zero.
(B) remains constant.
(C) continuously increases.
(D) continuously decreases.
vo
x
y
atvv o
tgvvyoy
gtθsinvv oy Eq 2
Constant acceleration
θsinvv oyo
Eq 1 θcosovxv
Constant velocity
Motion in Two Dimensions
Motion in Two Dimensions
A ball is thrown with a velocity of 20 m/s at an angle of 60° above the horizontal. What is the horizontal component of its instantaneous velocity at the exact top of its trajectory?
(A) 10 m/s
(B) 17 m/s
(C) 20 m/s
(D) zero
vo
x
y
SubEq 1
t cosvx o Eq 3
1 Eq. cosvv ox
tΔxΔ
v
tx
vx
tx
θcosvo
Constant velocity
Motion in Two Dimensions
Practice Problem
If Vx = 6.80 units and Vy = 7.40 units, a) determine the magnitude of V.
b) determine the direction of V
2y
2x
2 vvv
2y
2x vvv 22 7.406.80
Vx
V
Vy
units 10
x
y
v
vtan
x
y1
v
vtan
80.640.7
tan 1 o47
vo
x
y
2
gtt θsinvy
2
o Eq 4
2at
tvxx2
oo
Constant acceleration
2
tgtv0y
2
yo
2
θsin2tg
tvy o
θsinvv oyo
Motion in Two Dimensions
Motion in Two Dimensions
A soccer ball is kicked with a velocity of 25 m/s at an angle of 45° above the horizontal. What is the vertical component of its acceleration as it travels along its trajectory?
(A) 9.80 m/s2 downward
(B) (9.80 m/s2) × sin (45°) downward
(C) (9.80 m/s2) × sin (45°) upward
(D) (9.80 m/s2) upward
Motion in Two Dimensions
When a football in a field goal attempt reaches its maximum height, how does its speed compare to its initial speed?
(A) It is zero.
(B) It is equal to its initial speed.
(C) It is greater than its initial speed.
(D) It is less than its initial speed.
voh
x
y
Sub into Eq 4
θcosvx
to
2
ooo θ cosv
xg2
1θsinθ cosv
xvy
θcosv2
gxθtan xy
22o
2 Eq 5
3 Eq. t θcosvx o
4 Eq. 2
gtt θsinvy
2
o
Solve Eq 3 for t
Motion in Two Dimensions
Vertical Position as a Function of Horizontal Displacement
voh
x
y
At the maximum
height (vy = 0)
0gtθsinvv oy
Sub into Eq 4
2g
θsinv y
22o
max Eq 7
4 Eq. 2
gtt θsinvy
2
o
2 Eq.gt θsinvv oy
g
θsinvt o
top Eq 6
2
o
oo
g
θsinv
2
g
g
θsinv θsinvy
Motion in Two Dimensions
Maximum Height
Problem
A football is kicked at ground level with a speed of 18.0 m/s at an angle of 35.0º to the horizontal. How much later does it hit the ground?
g
sinvt otop
Time in the air is twice the time to the top.
g
sinv2t 2t otopair
2m/s 8.9
35sinm/s 182 s 1.2
vo
R
hx
y
Eq.6
g
θsinvt otop
g
θsinv2t2t otopair
airxtvR
g
θsinv2θcosvR o
o
g
θ2sinvR
2o Eq 8
1 Eq. θcosvv ox
Motion in Two Dimensions
Range
Motion in Two Dimensions
At what angle should a water-gun be aimed in order for the water to land with the greatest horizontal range?
(A) 0°
(B) 30°
(C) 45°
(D) 60°
The range of a projectile is maximum (if there is no air resistance) for a launch angle of 45°.
Motion in Two Dimensions
A projectile is fired with an initial speed of 65.2 m/s at an angle of 34.5º above the horizontal on a long flat firing range. Determine (a) the maximum height reached by the projectile.
Problem
g2
sinv y
22i
max
2
22
m/s 8.92
5.34sinm/s 2.65
(b) the total time in the air
g
sinv2t 2t otopair
2m/s 8.9
5.34sinm/s 5.262 s 5.7
(c) the total horizontal distance covered (that is, the range).
g
θ2sinvR
2o
2
o2
m/s 8.9
5.342sinm/s 2.65 m 405
m 6.69
You are trying to hit a friend with a water balloon. He is sitting in the window of his dorm room directly across the street. You aim straight at him and shoot. Just when you shoot, he falls out of the window! Does the water balloon hit him?
a) yes, it hits
b) maybe—it depends on the speed of the shot
c) no, it misses
d) the shot is impossible
e) not really sure
Assume that the shot does have enough speed to reach the dorm
across the street.
Your friend falls under the influence of gravity, just like the water balloon. Thus, they are both undergoing free fall in the y-direction. Since the slingshot was accurately aimed at the right height, the water balloon will fall exactly as your friend does, and it will hit him!!
Question 3.10a Shoot the Monkey I
Shoot the Monkey
cosvv ox Eq 1Horizontal Velocity
gtsinvv oy Eq 2Vertical Velocity
t cosvx o Eq 3Horizontal Displacement
2
gtt sinvy
2
o Eq 4Vertical Displacement
Equations
22
o
2
cosv2
gxtan xy Eq 5
Vertical Position
g2
sinv hy
22o
Eq 7Maximum Height
g
2sinvR
2o
Eq 8Range
g
sinvt otop
Eq 6Time to
the Top
Equations