Projectile Motion
KEY TERMS
Projectile - Object moving through the air, either initially thrown or dropped, subject only to the effects of gravity
Trajectory - The path of a projectile, which is parabolic in two dimensions
Projectile Motion - Movement of an object through the air, subject only to effects of gravity.
* In Projectile Motion, what happens in the vertical direction (y-direction) does NOT affect the horizontal direction (x-direction), and vice versa. An object's horizontal position, velocity, or acceleration does not affect its vertical position, velocity, or acceleration. These motions can only be related by the time variable t.
* Horizontal motion has constant velocity and zero acceleration while vertical motion has constant acceleration. This means for projectile motion, the starting velocity in the x-direction will be the same as the final velocity in the x-direction, while the starting and ending velocities in the y-direction will be different because of acceleration due to gravity.
Projectile Motion Graphs Analysis
Khan Academy video
KEY TERMS
Projectile - Object moving through the air, either initially thrown or dropped, subject only to the effects of gravity
Trajectory - The path of a projectile, which is parabolic in two dimensions
Projectile Motion - Movement of an object through the air, subject only to effects of gravity.
* In Projectile Motion, what happens in the vertical direction (y-direction) does NOT affect the horizontal direction (x-direction), and vice versa. An object's horizontal position, velocity, or acceleration does not affect its vertical position, velocity, or acceleration. These motions can only be related by the time variable t.
* Horizontal motion has constant velocity and zero acceleration while vertical motion has constant acceleration. This means for projectile motion, the starting velocity in the x-direction will be the same as the final velocity in the x-direction, while the starting and ending velocities in the y-direction will be different because of acceleration due to gravity.
Projectile Motion Graphs Analysis
Khan Academy video
Projectile Motion Equations
Vx --> velocity along x-axis,
Vxo --> initial velocity along x-axis,
Vy --> velocity along y-axis,
Vyo --> initial velocity along y-axis.
g --> acceleration due to gravity and
t --> time taken
Vx --> velocity along x-axis,
Vxo --> initial velocity along x-axis,
Vy --> velocity along y-axis,
Vyo --> initial velocity along y-axis.
g --> acceleration due to gravity and
t --> time taken
Additional Formulas
Sample Problems (in the file below)
Mastering Physics - Projectile Motion | |
File Size: | 1296 kb |
File Type: |
Uniform Circular Motion
KEY TERMS
Uniform Circular Motion - Motion in a circle at a constant speed
Angular Velocity - Measure of how an angle changes over time. The rotational analogue of linear velocity. Vector quantity with counterclockwise defined as the positive direction.
Centripetal acceleration - Acceleration pointed towards the center of a curved path and perpendicular to the object's velocity. Causes an object to change its direction and not its speed along a circular pathway.
Period - Time needed for one revolution. Inversely proportional to frequency.
Frequency - Number of revolutions per second for a rotating object.
Circular Motion Variables
𝚫𝜃 → angular displacement (in radians)
ω → angular velocity (in rad/sec)
𝜶 → angular acceleration (in rad/sec/sec)
KEY TERMS
Uniform Circular Motion - Motion in a circle at a constant speed
Angular Velocity - Measure of how an angle changes over time. The rotational analogue of linear velocity. Vector quantity with counterclockwise defined as the positive direction.
Centripetal acceleration - Acceleration pointed towards the center of a curved path and perpendicular to the object's velocity. Causes an object to change its direction and not its speed along a circular pathway.
Period - Time needed for one revolution. Inversely proportional to frequency.
Frequency - Number of revolutions per second for a rotating object.
Circular Motion Variables
𝚫𝜃 → angular displacement (in radians)
ω → angular velocity (in rad/sec)
𝜶 → angular acceleration (in rad/sec/sec)
T → period (in seconds)
𝑓 → frequency (in cycles/second or Hertz)
Circular Motion Equations
𝑓 → frequency (in cycles/second or Hertz)
Circular Motion Equations
Linear Velocity V.S. Angular Velocity
Angular velocity ω is angular displacement divided by time, while linear velocity 𝒗 is linear displacement divided by time (Figure 1). Linear velocity is also sometimes called tangential velocity.
*Why do we call it tangential velocity?
Angular Velocity & Linear Velocity V.S. Radius Angular velocity does not change with radius, but linear velocity does. For example, in a marching band line going around a corner, the person on the outside has to take the largest steps to keep in line with everyone else. Therefore, the outside person has a much larger linear velocity of every person in the line is the same because they are moving the same angle in the same amount of time (Figure 2). |
Sample Problem and Answer
Centripetal Acceleration & Centripetal Force
Centripetal Acceleration - Acceleration pointed towards the center of a curved path and perpendicular to the object's velocity. Causes an object to change its direction and not its speed along a circular pathway. Also called radial acceleration. SI units are m/s/s.
Centripetal Force - Net force acting in the direction towards the center of a circular path, causing centripetal acceleration. Direction is perpendicular to the object's linear velocity. Also sometimes called radial force.
Equations
Centripetal Acceleration - Acceleration pointed towards the center of a curved path and perpendicular to the object's velocity. Causes an object to change its direction and not its speed along a circular pathway. Also called radial acceleration. SI units are m/s/s.
Centripetal Force - Net force acting in the direction towards the center of a circular path, causing centripetal acceleration. Direction is perpendicular to the object's linear velocity. Also sometimes called radial force.
Equations
* It is important to understand that in uniform circular motion problems, centripetal force is also the net force acting on the object, therefore, the equations can be related and rearranged into many different forms according to serve in favor of the specific problem.
(Some other useful equations derived from the equations on the given equation sheet are on the right)
More about Centripetal Acceleration and Centripetal Force
Sample Problems (in the file below)
Mastering Physics - Uniform Circular Motion | |
File Size: | 470 kb |
File Type: |
Universal Law of Gravitation
Newton’s universal law of gravitation states that every particle in the universe attracts every other particle with a force along a line joining them.
* Gravitational attraction is along a line joining the centers of mass of these two bodies. The magnitude of the force is the same on each, consistent with Newton's third law.
For two bodies having masses M1 and M2 with a distance r between their centers of mass, the equation for Newton's universal law of gravitation is shown in Figure 3.
Since this equation tends to deal with huge objects, M1 & M2 are both measured in kilograms (kg), but is measured in meters (m) as in all other equations in this unit.
The most important number in this equation is G, the universal gravitational constant, which is always equal to 6.67 * 10-11 (N * m2)/(kg2)
* Note that the force of gravity between two objects is dependent on the masses of the objects and the distance between them. The force is always directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Why does the acceleration due to gravity g=9.81 m/s/s on Earth? How does it relate to the Universal Law of Gravitation?
Newton’s universal law of gravitation states that every particle in the universe attracts every other particle with a force along a line joining them.
* Gravitational attraction is along a line joining the centers of mass of these two bodies. The magnitude of the force is the same on each, consistent with Newton's third law.
For two bodies having masses M1 and M2 with a distance r between their centers of mass, the equation for Newton's universal law of gravitation is shown in Figure 3.
Since this equation tends to deal with huge objects, M1 & M2 are both measured in kilograms (kg), but is measured in meters (m) as in all other equations in this unit.
The most important number in this equation is G, the universal gravitational constant, which is always equal to 6.67 * 10-11 (N * m2)/(kg2)
* Note that the force of gravity between two objects is dependent on the masses of the objects and the distance between them. The force is always directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Why does the acceleration due to gravity g=9.81 m/s/s on Earth? How does it relate to the Universal Law of Gravitation?
HELPFUL VIDEO
Sample Problems (in the file below)
Mastering Physics - Universal Law of Gravitation | |
File Size: | 1124 kb |
File Type: |
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