Physics
Rest and Motion : Kinematics
Rest and Motion
11
⚡ Quick Summary
Rest and motion are relative. Something can be at rest to you, but moving to someone else.
N/A
Rest and motion are relative concepts. An object is said to be at rest if its position does not change with respect to a reference point. Otherwise, it is in motion.
Distance and Displacement
11
⚡ Quick Summary
Distance is how far you actually traveled. Displacement is how far you are from where you started (in a straight line).
N/A
Distance is the total length of the path traveled by an object. Displacement is the shortest distance between the initial and final positions of an object, along with its direction.
Average Speed and Instantaneous Speed
11
⚡ Quick Summary
Average speed is the total distance divided by the total time. Instantaneous speed is how fast you're going at a particular moment.
Average Speed = Total Distance / Total Time
Average speed is the total distance traveled divided by the total time taken. Instantaneous speed is the speed of an object at a specific instant in time.
Average Velocity and Instantaneous Velocity
11
⚡ Quick Summary
Average velocity is the total displacement divided by the total time. Instantaneous velocity is the velocity at a particular moment.
Average Velocity = Total Displacement / Total Time
Average velocity is the total displacement divided by the total time taken. Instantaneous velocity is the velocity of an object at a specific instant in time. It's a vector quantity, with both magnitude and direction.
Average Acceleration and Instantaneous Acceleration
11
⚡ Quick Summary
Average acceleration is the change in velocity divided by the change in time. Instantaneous acceleration is how quickly your velocity is changing at a particular moment.
Average Acceleration = (Change in Velocity) / (Change in Time)
Average acceleration is the change in velocity divided by the change in time. Instantaneous acceleration is the acceleration of an object at a specific instant in time.
Motion in a Straight Line
11
⚡ Quick Summary
Motion in one dimension. We can use equations to describe the position, velocity, and acceleration of an object moving along a line.
v = u + at, s = ut + (1/2)at^2, v^2 = u^2 + 2as
Motion in a straight line involves displacement, velocity, and acceleration along a single axis. Constant acceleration equations are commonly used to analyze this type of motion.
Motion in a Plane
11
⚡ Quick Summary
Motion in two dimensions (like throwing a ball). We need to consider both x and y components.
N/A
Motion in a plane involves displacement, velocity, and acceleration in two dimensions. It can be analyzed by resolving the motion into its x and y components.
Projectile Motion
11
⚡ Quick Summary
Projectile motion is when you throw something and it follows a curved path. Gravity pulls it down!
Range = (u^2 * sin(2θ)) / g, Max Height = (u^2 * sin^2(θ)) / (2g)
Projectile motion is the motion of an object launched into the air and subject to gravity. The trajectory is a parabola. We often ignore air resistance for simplicity.
Change of Frame
11
⚡ Quick Summary
Changing your point of view can make a problem easier. Imagine watching a ball rolling on a train – it looks different from the train than it does from the ground!
N/A
A frame of reference is a coordinate system used to describe the motion of an object. Changing the frame of reference can simplify the analysis of motion.
Distance travelled from speed-time graph
11
⚡ Quick Summary
The distance travelled by a particle in a given time interval can be found by calculating the area under the speed-time graph for that time interval.
s = ∫v dt (where the integral is taken from t1 to t2)
If you have a graph showing how the speed (v) of an object changes over time (t), the distance it travels is equal to the area between the curve of the graph and the x-axis (time axis). Imagine dividing the area into tiny rectangles; each rectangle's area is approximately the distance traveled in that small time interval. Adding up all these tiny distances gives you the total distance.
Average Velocity
11
⚡ Quick Summary
Average velocity is how much an object's position changes (displacement) over a certain amount of time.
v_av = (AB) / (t2 - t1)
Average velocity is a vector quantity, meaning it has both magnitude and direction. It's calculated by dividing the displacement (change in position) by the time interval. Displacement is the straight-line distance and direction between the starting and ending points, regardless of the actual path taken.
Position Vector
11
⚡ Quick Summary
A position vector points from the origin of a coordinate system to the location of an object.
AB = r2 - r1
The position vector is a vector that starts at the origin (0,0,0) of a coordinate system and ends at the current location of the object. It tells you exactly where the object is in space relative to your reference point (the origin).
Relative Velocity
11
⚡ Quick Summary
The velocity of an object as seen from another moving object. Imagine you're in a car, and you see another car moving. How fast it *seems* to be going depends on how fast *you're* going too!
v_rain,man = v_rain,street - v_man,street
The velocity of rain with respect to a man is given by: v_rain,man = v_rain,street - v_man,street where v_rain,street is the velocity of the rain with respect to the street and v_man,street is the velocity of the man with respect to the street.
Relative Acceleration
11
⚡ Quick Summary
How the acceleration of an object changes when viewed from a different accelerating frame of reference. If you're in a car accelerating, things outside might *seem* to be accelerating differently than if you were standing still.
a_P,S = a_P,S' + a_S',S
The relation between accelerations measured from two frames is: a_P,S = a_P,S' + a_S',S where a_P,S is the acceleration of particle P with respect to frame S, a_P,S' is the acceleration of particle P with respect to frame S', and a_S',S is the acceleration of frame S' with respect to frame S. If S' moves with respect to S at a uniform velocity, a_S',S = 0 and thus a_P,S = a_P,S'.
Equation of Motion with Constant Acceleration
Class 11
⚡ Quick Summary
This describes the distance covered by an object undergoing constant acceleration over a period of time.
s = ut + (1/2)at^2
This equation relates the distance (s) traveled to the initial velocity (v), time (t), and constant acceleration (a).
Acceleration in Frames Moving with Uniform Velocity
11
⚡ Quick Summary
If two observers are moving at a constant speed relative to each other, they will measure the same acceleration for any object they observe.
a_P,S = a_P,S' (when relative velocity between frames S and S' is constant)
If two frames are moving with respect to each other with uniform velocity, the acceleration of a body is the same in both frames. This is because the relative acceleration between the frames is zero.
Time of Flight with Constant Acceleration
Class 11
⚡ Quick Summary
Deals with finding the time at which a pickpocket will be caught by a jeep, based on the jeep's acceleration (a), the initial relative velocity (v), and the initial distance (d).
t = (v – √(v^2 - 2ad))/a
The pickpocket will be caught if the time (t) is real and positive. This condition is met when v^2 >= 2ad
Change in Velocity
Class 11
⚡ Quick Summary
The change in velocity is the difference between the final and initial velocities. Even if the speed stays the same, a change in direction means there's a change in velocity!
Δv = v_final - v_initial
Change in velocity is a vector quantity, considering both magnitude and direction. A change in direction constitutes a change in velocity even if the speed remains constant.
Displacement-Time Graph
Class 11
⚡ Quick Summary
A displacement-time graph shows where an object is at different times. The slope of the graph tells you the object's velocity.
Velocity = Δx/Δt (where Δx is change in displacement and Δt is change in time)
A displacement-time graph plots the position of an object against time. The slope of the graph at any point represents the instantaneous velocity of the object.
Average Velocity
Class 11
⚡ Quick Summary
Average velocity is the total displacement divided by the total time. It's like saying, 'If the object moved at a constant speed the whole time, what speed would it need to have to travel the same distance?'
Average velocity = Total displacement / Total time
Average velocity considers the total displacement (change in position) and the total time taken, regardless of the variations in velocity during the motion. It is a vector quantity.
Acceleration due to Gravity
Class 11
⚡ Quick Summary
When something is falling (and we ignore air resistance), it speeds up at a constant rate because of gravity. This rate is about 9.8 m/s every second.
g ≈ 9.8 m/s²
The acceleration due to gravity (g) is approximately 9.8 m/s² near the Earth's surface. It acts downwards. When an object is released, gravity is the main force to consider.
Relative Motion
Class 11
⚡ Quick Summary
How things look depend on your point of view. If you're moving, things around you seem to move differently than if you were standing still!
v_AB = v_A - v_B (Velocity of A with respect to B)
Relative motion describes how the motion of an object appears from different frames of reference (i.e., different moving observers).
Projectile Motion
11
⚡ Quick Summary
Projectile motion is when an object is thrown or launched into the air and follows a curved path due to gravity. Understanding this motion involves analyzing the horizontal and vertical components of its velocity and displacement.
Range (R) = (u^2 * sin(2θ))/g , Max Height (H) = (u^2 * sin^2(θ))/(2g) , Time of Flight (T) = (2u * sin(θ))/g where u is the initial velocity, θ is the angle of projection, and g is the acceleration due to gravity.
- Projectile motion is a combination of uniform motion in the horizontal direction and uniformly accelerated motion in the vertical direction (due to gravity).
- The horizontal component of velocity remains constant throughout the motion (if air resistance is negligible).
- The vertical component of velocity changes due to gravity.
Projectile Motion - Angle of Projection
11
⚡ Quick Summary
When you throw something at an angle, the best angle depends on what you're trying to do! Sometimes a higher angle is better, sometimes lower.
N/A (The text doesn't explicitly give formulas, but implies relationships between angle, range, and height)
The text implicitly refers to the concept that different angles of projection result in different ranges and maximum heights in projectile motion. There are optimal angles depending on the goal (maximum range vs. maximum height). It is also states that an angle of 5 degrees is not good, 15 might be OK, and 75 is not optimal.
Relative Motion - Angle and Time to Cross a Path
11
⚡ Quick Summary
When something is moving, and you're also moving (like a boat crossing a river), the direction you need to aim isn't straight across. It depends on how fast the other thing is moving and how fast you are!
N/A (Implies vector addition of velocities, but no formulas given)
The text alludes to problems involving finding the angle (direction) and time needed for an object to cross a moving medium, as in a boat crossing a river with a current. The angle depends on relative velocities of the boat and the river.
Projectile Motion - Minimum Speed for a Successful Shot
11
⚡ Quick Summary
How fast you need to throw something to hit a target depends on how far away it is. If you don't throw it fast enough, it won't make it!
v^2 - u^2 (This is an incomplete formula, but suggests the use of kinematic equations to find minimum velocity)
This refers to the minimum initial velocity required for a projectile to reach a certain point. This is implicitly linked to projectile motion formulas concerning range and height, and involves considering the gravitational force.
Motion with Constant Acceleration - Time to Reach a Certain Velocity
11
⚡ Quick Summary
If something is speeding up steadily, the time it takes to reach a specific speed depends on how quickly it's accelerating!
2 a/v (Incomplete formula, should be related to kinematic equations like v = u + at)
In uniform acceleration, the time taken to reach velocity 'v' from initial velocity 'u' depends on the acceleration 'a'.