AP Physics 1 projectile motion cheat sheet

What defines projectile motion?

Motion of an object thrown into the air under the influence of gravity alone. - Trajectory is a parabola. - Horizontal motion is uniform. - Vertical motion is uniformly accelerated.

Components of projectile motion?

Two independent components: 1. Horizontal (constant velocity) 2. Vertical (accelerated due to gravity, $\displaystyle g = 9.81 \, m/s^2$)

True or False: Vertical and horizontal motions are dependent.

False. They are independent of each other.

What is the effect of air resistance?

Air resistance reduces the range and alters the trajectory, making it less than ideal projectile motion.

How does gravity affect a projectile?

Gravity acts downward, causing: - Vertical acceleration - Constant downward velocity change - Maximum height reached when vertical velocity is zero.

Fill in the blank: The time of flight depends on _____.

Initial vertical velocity and height. Calculated using $\displaystyle t = \\frac{2v_{iy}}{g}$ for symmetrical paths.

What is maximum height?

The peak point of the projectile's trajectory, where vertical velocity is zero. Formula: $\displaystyle H = \\frac{v_{iy}^2}{2g}$.

Comparison: Range vs. Height

Range is the horizontal distance traveled; height is the vertical distance from launch to peak.

Define launch angle.

The angle between the initial velocity vector and the horizontal. Influences both range and maximum height.

How to calculate range?

Use the formula: $\displaystyle R = \\frac{v_{0}^2 \sin(2\theta)}{g}$, where $\displaystyle heta$ is the launch angle.

True or False: All projectiles have the same range.

False. Range varies with launch angle and initial speed.

What happens at peak height?

Vertical velocity is zero; only horizontal motion continues. Total time to peak is $\displaystyle t_{peak} = \\frac{v_{iy}}{g}$.

Effect of increasing launch speed?

Increases both range and maximum height. Range formula directly depends on $\displaystyle v_{0}^2$.

Define trajectory.

The path followed by a projectile, typically parabolic in shape due to constant gravity.

What is the horizontal velocity?

Remains constant throughout the motion since no horizontal acceleration acts on the projectile.

Real life example of projectile motion?

A basketball shot, where the ball follows a parabolic path influenced by launch speed and angle.

What is the formula for horizontal displacement?

Horizontal displacement ($\displaystyle ext{Range}$): $\displaystyle R = v_{0x} imes t$, where $\displaystyle v_{0x} = v_0 imes ext{cos}( heta)$.

Fill in the blank: The vertical motion is influenced by __________.

Gravity ($\displaystyle g = 9.81 ext{ m/s}^2$).

True or False: Time of flight is the same for all launch angles.

False. Time of flight varies with launch angle and initial velocity.

What is the equation for vertical displacement?

Vertical displacement: $\displaystyle y = v_{0y} imes t - \frac{1}{2} g t^2$, where $\displaystyle v_{0y} = v_0 imes ext{sin}( heta)$.

Cause → Effect: Increasing launch speed affects range how?

Increases range due to greater horizontal displacement.

Which variable represents initial velocity in projectile motion?

$\displaystyle v_0$ (initial velocity magnitude).

What is the formula for maximum height?

Maximum height ($\displaystyle H$): $\displaystyle H = \frac{v_{0y}^2}{2g}$. Note: $\displaystyle v_{0y} = v_0 imes ext{sin}( heta)$.

Comparison: Range vs. Maximum Height.

Range is horizontal distance; Maximum Height is the peak vertical position.

What is the time of flight formula?

Time of flight: $\displaystyle t = \frac{2v_{0y}}{g}$. $\displaystyle v_{0y} = v_0 imes ext{sin}( heta)$.

Fill in the blank: The vertical component of velocity at peak is __________.

Zero (0 m/s).

True or False: The horizontal velocity changes during flight.

False. Horizontal velocity remains constant (ignoring air resistance).

What are the components of initial velocity?

Horizontal: $\displaystyle v_{0x} = v_0 imes ext{cos}( heta)$; Vertical: $\displaystyle v_{0y} = v_0 imes ext{sin}( heta)$.

What is the total time in the air for a projectile launched at 45°?

Use $\displaystyle t = \frac{2v_{0} imes ext{sin}(45°)}{g}$. Simplifies to $\displaystyle t = \frac{v_0 ext{sqrt}(2)}{g}$.

Cause → Effect: Increasing launch angle effects range how?

Increases up to 45°, then decreases.

What is the relationship between angle and range?

Range is maximized at 45° launch angle (assuming no air resistance).

Vertical velocity formula at any time t?

$\displaystyle v_y = v_{0y} - gt$.

What is the formula for final vertical velocity?

Final vertical velocity: $\displaystyle v_y = v_{0y} - gt$, where $\displaystyle t$ is the time.

What does $\displaystyle g$ represent?

Acceleration due to gravity, approximately $\displaystyle 9.81 ext{ m/s}^2$.

Fill in the blank: The flight path of a projectile is __________.

A parabola.

What is the formula for horizontal motion?

The horizontal motion can be described by the equation: $$x = v_{0x} imes t$$ where: - $\displaystyle x$ = horizontal displacement - $\displaystyle v_{0x}$ = initial horizontal velocity - $\displaystyle t$ = time of flight.

What is the ideal launch angle for maximum range?

45 degrees is the ideal launch angle for maximum range in projectile motion.

True or false: Launch angle affects maximum height and range.

True - A higher launch angle increases maximum height but decreases range.

Fill in the blank: For a given speed, the maximum range occurs at __________.

45 degrees.

Compare range at 30° vs. 60°.

Both yield the same range, but 30° has a higher horizontal velocity, while 60° has a higher vertical component.

What happens to range as launch angle decreases from 45°?

Range decreases after reaching 45 degrees, as more energy is vertical than horizontal.

Cause → Effect: Increasing launch speed impacts __________.

Range and maximum height increase.

True or false: The time of flight is the same for angles summing to 90°.

True - For angles like 30° and 60°, time of flight remains equal.

Derive the range formula: R = ?

$\displaystyle R = \\frac{v_0^2 \sin(2\theta)}{g}$, where $\displaystyle v_0$ is initial velocity and $\displaystyle g$ is acceleration due to gravity.

Highest point in trajectory occurs at __________.

Half the total time of flight.

What is the effect of air resistance on range?

Air resistance reduces range compared to ideal conditions.

Determine maximum height formula: H = ?

$\displaystyle H = \\frac{v_0^2 \sin^2(\theta)}{2g}$.

Impact of launch angle on maximum height: Higher angles result in __________.

Higher maximum heights.

What is the relationship between horizontal velocity and launch angle?

Horizontal velocity decreases as launch angle increases, while vertical velocity increases.

Fill in the blank: The range is maximized when the launch angle is __________ and the speed is constant.

45 degrees.

Calculate range for a 20 m/s launch at 30°: R = ?

$\displaystyle R = \\frac{(20)^2 \sin(60)}{9.81} \approx 40.8 m$.

True or false: Maximum height is reached at 90° launch angle.

True - A 90° angle achieves maximum height but no horizontal range.

Baseball thrown at an angle →

It follows a parabolic path. Maximum height and distance depend on launch speed and angle.

True or False: A dropped ball and a ball thrown horizontally hit the ground at the same time.

True. Both are subject to the same gravitational acceleration (ignoring air resistance).

Fill in the blank: The range of a projectile is maximized at _____ degrees.

45 degrees.

Example: Calculating range of a soccer ball kicked at 30 m/s.

Use the formula: $\displaystyle R = \\frac{v^2 \sin(2\theta)}{g}$. For $\displaystyle \theta = 45^\circ$, $\displaystyle R \approx 90 m$.

Effect of air resistance on projectiles →

Reduces range and maximum height. Affects lighter projectiles more significantly.

Comparison: Horizontal vs. Vertical motion of a projectile.

Horizontal: constant velocity; Vertical: accelerated due to gravity.

True or False: The time of flight is independent of horizontal distance.

True. Time of flight only depends on the initial vertical velocity and height.

Real-world application: Water fountains use projectile motion to _____

Demonstrate parabolic trajectories; optimize height and distance for aesthetic effects.

Question: How does angle affect a basketball shot's success?

Higher angles increase maximum height, improving chances of successful shots.

Cause → Effect: Increasing launch speed on a projectile.

Increases range and maximum height achieved.

Example: Golf ball's trajectory on a tee shot.

Kicked at an angle, the ball follows a parabolic path, maximizing distance at optimal angle.

Real-world application: Archery targets are set at specific distances due to _____

Predicted projectile motion. Accurate range calculations ensure hits on target.