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Scalar
Physical quantities that have a magnitude, but no direction. Mass and temperature are examples.
Magnitude
Numerical value of a scalar or vector.
Vector
Physical quantities that have both a magnitude and a direction. Force and acceleration are examples.
Position
Where an object is located. It is designated by x. When an object moves, the position changes, which is designated by Δx. This can be found by taking the integral of velocity with respect to time. In free fall, this is designated by Δy.
Distance
Total length of travel from initial to final position, and it is a scalar quantity.
Displacement
Straight-line distance from initial to final position, and it is a vector quantity.
Meter (m)
Standard unit of measurement for distance or displacement.
Speed
Distance divided by time. Scalar quantity. Designated by v.
Velocity
Displacement divided by time. Vector quantity. Designated by v. It is the first derivative of position with respect to time. It is the integral of acceleration with respect to time.
Position vs time graph
Object’s position is read from the vertical axis. The slope of the graph is the object’s velocity. The steeper the slope, either positive or negative, the faster the object moves. If the slope is zero, the object is immobile. If the slope is positive, the object is moving in a positive direction, and if the slope is negative, the object is moving in a negative direction. If the graph is concave up, the object is accelerating in the positive direction. If the graph is concave down, the object is accelerating in the negative direction.
Velocity vs time graph
Object’s velocity is read from the vertical axis. Direction of motion indicated by the sign of the velocity. The area between the graph and the horizontal axis is the object’s displacement. The position of the object can not be determined from this graph, only how far the object is from its starting point. If the slope is zero, the object is either immobile or moving at a constant velocity. If the slope is positive, the object is accelerating in the positive direction. If the slope is negative, the object is accelerating in the negative direction.
Relative Velocity
Velocity of a moving object as measured by another moving object. When two separate objects are moving in opposite directions, this is found by adding the two speeds. When two separate objects are moving in the same direction, this is found by subtracting the two speeds. When one object is moving in or on another object moving in the same direction, this can be found by adding the two speeds. When one object is moving in or on another object moving in the opposite direction, this can be found by subtracting the two speeds.
Average Speed
Total distance divided by total time. Designated by Vavg. If an object moves at two different speeds for the same time, this can be found by averaging the two speeds. However, if the object moves at two different speeds for the same distance, this cannot be found by averaging the two speeds. The object spends more time moving at a slower speed, lowering this.
Instantaneous Speed
Speed of an object at any given instant in time. It can be found by taking the limit of the speed as the time interval approaches zero.
Acceleration
Measure of how quickly velocity changes (change in magnitude or direction.) Designated by the letter a. For constant acceleration, it is equal to the change in velocity (V-Vo) divided by t, time. It is the first derivative of velocity with respect to time. It is measured in meters per second squared (m/s^2). It is a vector. If an object is speeding up, this is in the same direction as the object’s velocity. If an object is slowing down, this is in the opposite direction of the object’s velocity. If an object changes direction while moving at a constant speed, this is perpendicular to the direction of the velocity (centripetal this.)
Instantaneous acceleration
Acceleration at any given point in time. Given by finding the limit of the acceleration as the time interval approaches zero.
Kinematics Equation (missing Δx)
V = Vo + at
Kinematics Equation (missing a)
Δx = (1/2)(Vo+V)(t)
Kinematics Equation (missing V)
Δx = Vot + 1/2at^2
Kinematics Equation (missing Vo)
Δx = Vt - 1/2at^2
Kinematics Equation (missing t)
V^2 = Vo^2 + 2aΔx
Acceleration vs. time graph
Object’s acceleration is read from the vertical axis. If there is a horizontal line at zero, the object is either not moving or moving at a constant velocity. If there is a horizontal line above or below zero, the object is accelerating in the positive or negative directions, respectively. If the graph has a slope or concavity, the acceleration is changing.
Free Fall
Occurs when the only force acting upon an object is the gravitational force. This can occur for objects that are rising or falling through the air, “fall” does not necessarily mean down.
Gravitational Force
Gravity causes objects on Earth to accelerate downward at a rate of 9.8 m/s^2. On labs, 9.8 m/s^2 should be used, but on tests and other assignments, 10 m/s^2 can be used. For rising objects, the speed decreases by 10 m/s each second. For falling objects, the speed increases by 10 m/s each second. Since gravitational pull is constant, the kinematics equations can be used. Air resistance acts on objects when they are in free fall, but right now, it should be considered negligible.
Five main principles for objects launched straight up that come straight back down
When the object is first launched, Vo is not equal to 0
If the object starts and ends at the same point, the time going up is equal to the time going down is equal to half the total time in motion
At any particular height, Vup = -Vdown
At the very top of the path, V = 0
At all points along the path, a = -g
Projectile Motion
Special case of free fall where an object follows a 2-dimensional trajectory. It occurs when an object is either launched horizontally or at an angle above or below the horizontal. Every point on the object’s trajectory has a velocity with x and y components. In vector form, this looks like r = xi + yj + zk. To solve the problem, the motion must be broken down into those components. Acceleration in the x direction is always zero, so the x velocity stays constant and the graph of x as a function of time is linear. Acceleration in the y direction is always the force of gravity, so the y velocity decreases at a constant rate, and therefore y as a function of time is quadratic. The equation V = Δx/t should be used to solve the horizontal motion, while the kinematics equations should be used to solve the vertical motion. Equations for both the x and y position can be found by rearranging the above equations and plugging in the known and unknown information. By eliminating the parameter, the equation for the parabolic function can be found. An arc length formula can be used to find the distance traveled as well. When there is no air resistance, maximum range occurs when an object is launched at 45° angle. Complimentary angles have the same range.
Horizontal velocity for an object launched at an angle θ above the horizontal
Vox = Vocosθ
Vertical velocity for an object launched at an angle θ above the horizontal
Voy = Vosinθ
Motion Diagram/Motion Map
Visual representation of an object showing its position and velocity at equal time intervals. Position is represented by a dot, and velocity is represented by an arrow pointing at the next dot. When moving at a constant velocity, the dots are equally spaced, and the arrows have equal length. The farther apart the dots are, the faster the object is moving. Longer arrow length also represents faster motion. When an object is stationary, the dots should simply be stacked on top of each other.
Frame of Reference
Point of view from which one observes motion. Usually, this does not move, but if it is moving, like if it is inside an elevator or vehicle, relative velocity must be considered.
Inertial reference frame
Reference frame that is not moving or moving with a constant velocity. They are all equivalent, and the acceleration of an object is constant.
Non-inertial reference frame
Reference frame that is accelerating. The acceleration would be different than the theoretical acceleration.
Maximum/Minimum values of Position, Velocity, and Acceleration
Set the first derivative equal to zero to find the critical values. Check the second derivative at each of these critical values to find whether it is a minimum or maximum. In addition, check the endpoints of the original function.
Note
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