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Chapter 4
Two-Dimensional Kinematics
Units of Chapter 4
• Motion in Two Dimensions
• Projectile Motion: Basic Equations
• Zero Launch Angle
• General Launch Angle
• Projectile Motion: Key Characteristics
4-1 Motion in Two Dimensions – No Acceleration
If velocity is constant, motion is along a straight line:
4-1 Motion in Two Dimensions – Constant Acceleration
Motion in the x- and y-directions should be solved separately:
ConcepTest 4.1aConcepTest 4.1a Firing Balls IFiring Balls I
A small cart is rolling at
constant velocity on a flat
track. It fires a ball straight
up into the air as it moves.
After it is fired, what happens
to the ball?
1) it depends on how fast the cart is 1) it depends on how fast the cart is movingmoving
2) it falls behind the cart2) it falls behind the cart
3) it falls in front of the cart3) it falls in front of the cart
4) it falls right back into the cart4) it falls right back into the cart
5) it remains at rest5) it remains at rest
ConcepTest 4.1aConcepTest 4.1a Firing Balls IFiring Balls I
A small cart is rolling at
constant velocity on a flat
track. It fires a ball straight
up into the air as it moves.
After it is fired, what happens
to the ball?
1) it depends on how fast the cart is 1) it depends on how fast the cart is movingmoving
2) it falls behind the cart2) it falls behind the cart
3) it falls in front of the cart3) it falls in front of the cart
4) it falls right back into the cart4) it falls right back into the cart
5) it remains at rest5) it remains at rest
when viewed from
train
when viewed from
ground
In the frame of reference of the cart, the ball only has a verticalvertical component of velocity. So it goes up and comes back down. To a ground observer, both the cart and the ball have the same horizontal velocitysame horizontal velocity, so the ball still returns into the cart.
4-2 Projectile Motion: Basic Equations
Assumptions:
• ignore air resistance
• g = 9.81 m/s2, downward
• ignore Earth’s rotation
If y-axis points upward, acceleration in x-direction is zero and acceleration in y-direction is -9.81 m/s2
4-2 Projectile Motion: Basic Equations
The acceleration is independent of the direction of the velocity:
4-2 Projectile Motion: Basic Equations
These, then, are the basic equations of projectile motion:
4-3 Zero Launch Angle
Launch angle: direction of initial velocity with respect to horizontal
4-3 Zero Launch Angle
In this case, the initial velocity in the y-direction is zero. Here are the equations of motion, with x0 = 0 and y0 = h:
ConcepTest 4.2ConcepTest 4.2 Dropping a PackageDropping a Package
You drop a package from You drop a package from
a plane flying at constant a plane flying at constant
speed in a straight line. speed in a straight line.
Without air resistance, the Without air resistance, the
package will:package will:
1) quickly lag behind the plane quickly lag behind the plane while fallingwhile falling
2) remain vertically under the 2) remain vertically under the plane while fallingplane while falling
3) move ahead of the plane while 3) move ahead of the plane while fallingfalling
4) not fall at all4) not fall at all
You drop a package from You drop a package from
a plane flying at constant a plane flying at constant
speed in a straight line. speed in a straight line.
Without air resistance, the Without air resistance, the
package will:package will:
1) quickly lag behind the plane quickly lag behind the plane while fallingwhile falling
2) remain vertically under the 2) remain vertically under the plane while fallingplane while falling
3) move ahead of the plane while 3) move ahead of the plane while fallingfalling
4) not fall at all4) not fall at all
Both the plane and the package have the
samesame horizontalhorizontal velocityvelocity at the moment of
release. They will maintainmaintain this velocity in
the xx-direction-direction, so they stay aligned.
Follow-up:Follow-up: What would happen if air resistance is present? What would happen if air resistance is present?
ConcepTest 4.2ConcepTest 4.2 Dropping a PackageDropping a Package
ConcepTest 4.3aConcepTest 4.3a Dropping the Ball IDropping the Ball I
From the same height (and
at the same time), one ball
is dropped and another ball
is fired horizontally. Which
one will hit the ground
first?
(1) the “dropped” ball(1) the “dropped” ball
(2) the “fired” ball(2) the “fired” ball
(3) they both hit at the same time(3) they both hit at the same time
(4) it depends on how hard the ball (4) it depends on how hard the ball was firedwas fired
(5) it depends on the initial height(5) it depends on the initial height
From the same height (and
at the same time), one ball
is dropped and another ball
is fired horizontally. Which
one will hit the ground
first?
(1) the “dropped” ball(1) the “dropped” ball
(2) the “fired” ball(2) the “fired” ball
(3) they both hit at the same time(3) they both hit at the same time
(4) it depends on how hard the ball (4) it depends on how hard the ball was firedwas fired
(5) it depends on the initial height(5) it depends on the initial height
Both of the balls are falling vertically under the influence of
gravity. They both fall from the same height.They both fall from the same height. Therefore, they will Therefore, they will
hit the ground at the same time.hit the ground at the same time. The fact that one is moving
horizontally is irrelevant – remember that the x and y motions are
completely independent !!
Follow-up:Follow-up: Is that also true if there is air resistance? Is that also true if there is air resistance?
ConcepTest 4.3aConcepTest 4.3a Dropping the Ball IDropping the Ball I
ConcepTest 4.3bConcepTest 4.3b Dropping the Ball IIDropping the Ball II
In the previous problem, In the previous problem,
which ball has the greater which ball has the greater
velocity at ground level?velocity at ground level?
1) the “dropped” ball1) the “dropped” ball
2) the “fired” ball2) the “fired” ball
3) neither – they both have the 3) neither – they both have the
same velocity on impactsame velocity on impact
4) it depends on how hard the 4) it depends on how hard the
ball was thrownball was thrown
In the previous problem, In the previous problem,
which ball has the greater which ball has the greater
velocity at ground level?velocity at ground level?
1) the “dropped” ball1) the “dropped” ball
2) the “fired” ball2) the “fired” ball
3) neither – they both have the 3) neither – they both have the
same velocity on impactsame velocity on impact
4) it depends on how hard the 4) it depends on how hard the
ball was thrownball was thrown
Both balls have the same same verticalvertical velocity velocity when they hit the ground (since they are both acted on by gravity for the same time). However, the “fired” ball also has a horizontal velocity. When you add the two components vectorially, the “fired” ball has “fired” ball has a larger net velocitya larger net velocity when it hits the ground.
ConcepTest 4.3bConcepTest 4.3b Dropping the Ball IIDropping the Ball II
4-4 General Launch Angle
In general, v0x = v0 cos θ and v0y = v0 sin θ
This gives the equations of motion:
4-4 General Launch Angle
Snapshots of a trajectory; red dots are at t = 1 s, t = 2 s, and t = 3 s
4-5 Projectile Motion: Key Characteristics
Range: the horizontal distance a projectile travels
If the initial and final elevation are the same:
4-5 Projectile Motion: Key Characteristics
The range is a maximum when θ = 45°:
Symmetry in projectile motion:
4-5 Projectile Motion: Key Characteristics
Which of the Which of the
3 punts has 3 punts has
the longest the longest
hang time?hang time?
ConcepTest 4.4aConcepTest 4.4a Punts IPunts I
4) all have the same hang time
1 2 3
h
Which of the Which of the
3 punts has 3 punts has
the longest the longest
hang time?hang time?
4) all have the same hang time
1 2 3
h
The time in the air is determined by the vertical vertical
motion motion ! Since all of the punts reach the same heightsame height,
they all stay in the air for the same timesame time.
Follow-up:Follow-up: Which one had the greater initial velocity? Which one had the greater initial velocity?
ConcepTest 4.4aConcepTest 4.4a Punts IPunts I
ConcepTest 4.4bConcepTest 4.4b Punts IIPunts II
1 2
3) both at the same time
A battleship simultaneously fires two shells at two enemy A battleship simultaneously fires two shells at two enemy
submarines. The shells are launched with the submarines. The shells are launched with the samesame initial initial
velocity. If the shells follow the trajectories shown, which velocity. If the shells follow the trajectories shown, which
submarine gets hit submarine gets hit firstfirst ? ?
1 2
3) both at the same time
A battleship simultaneously fires two shells at two enemy A battleship simultaneously fires two shells at two enemy
submarines. The shells are launched with the submarines. The shells are launched with the samesame initial initial
velocity. If the shells follow the trajectories shown, which velocity. If the shells follow the trajectories shown, which
submarine gets hit submarine gets hit firstfirst ? ?
The flight time is fixed by the motion in the y-direction. The higher an object goes, the longer it stays in flight. The shell hitting submarine #2 goes less high, therefore it stays in flight for less time than the other shell. Thus,
submarine #2 is hit first.
Follow-up:Follow-up: Which one traveled the greater distance? Which one traveled the greater distance?
ConcepTest 4.4bConcepTest 4.4b Punts IIPunts II
The spring-loaded gun can launch
projectiles at different angles with the
same launch speed. At what angle
should the projectile be launched in
order to travel the greatest distance
before landing?
1) 15°
2) 30°
3) 45°
4) 60°
5) 75°
ConcepTest 4.6ConcepTest 4.6 Spring-Loaded GunSpring-Loaded Gun
The spring-loaded gun can launch
projectiles at different angles with the
same launch speed. At what angle
should the projectile be launched in
order to travel the greatest distance
before landing?
1) 15°
2) 30°
3) 45°
4) 60°
5) 75°
A steeper angle lets the projectile stay in the air longer, but it does not travel so far because it has a small x-component of velocity. On the other hand, a shallow angle gives a large x-velocity, but the projectile is not in the air for very long. The compromise comes at 45°, although this result is best seen in a calculation of the “range formula” as shown in the textbook.
ConcepTest 4.6ConcepTest 4.6 Spring-Loaded GunSpring-Loaded Gun
Example
• A golfer hits a shot with an initial speed of 42 m/s at an angle of 35° above the vertical. Neglect air resistance
1. What maximum height does the ball reach?2. How long does it take to get there?3. How long does it take to get back down?4. How far does it travel horizontally?5. With what velocity (magnitude and direction)
will it hit the ground, if it lands at the same elevation it started from?
Summary of Chapter 2
• Distance: total length of travel
• Displacement: change in position
• Average speed: distance / time
• Average velocity: displacement / time
• Instantaneous velocity: average velocity measured over an infinitesimally small time
Summary of Chapter 2
• Instantaneous acceleration: average acceleration measured over an infinitesimally small time
• Average acceleration: change in velocity divided by change in time
• Deceleration: velocity and acceleration have opposite signs
• Constant acceleration: equations of motion relate position, velocity, acceleration, and time
• Freely falling objects: constant acceleration g = 9.81 m/s2
Summary of Chapter 3
• Scalar: number, with appropriate units
• Vector: quantity with magnitude and direction
• Vector components: Ax = A cos θ, By = B sin θ
• Magnitude: A = (Ax2 + Ay
2)1/2
• Direction: θ = tan-1 (Ay / Ax)
• Graphical vector addition: Place tail of second at head of first; sum points from tail of first to head of last
• Component method: add components of individual vectors, then find magnitude and direction• Unit vectors are dimensionless and of unit length• Position vector points from origin to location• Displacement vector points from original position to final position• Velocity vector points in direction of motion• Acceleration vector points in direction of change of motion• Relative motion: v13 = v12 + v23
Summary of Chapter 3
Summary of Chapter 4
• Components of motion in the x- and y-directions can be treated independently
• In projectile motion, the acceleration is –g
• If the launch angle is zero, the initial velocity has only an x-component
• The path followed by a projectile is a parabola
• The range is the horizontal distance the projectile travels
