Physics
Motion in Two Dimensions
Average Velocity
11
⚡ Quick Summary
Average velocity is the displacement (change in position) divided by the time it took. It only cares about where you started and where you ended up, not the path you took in between.
v_av = (r2 - r1) / (t2 - t1)
- Average velocity (v_av) is calculated over a time interval Δt.
- v_av = (r2 - r1) / (t2 - t1), where r1 and r2 are the positions at times t1 and t2, respectively.
- The actual path taken between the initial and final positions is not considered.
Instantaneous Velocity
11
⚡ Quick Summary
Instantaneous velocity is how fast and in what direction you're moving at a specific moment in time. Imagine looking at the speedometer of a car at one particular instant.
v = lim (Δt→0) (Δr / Δt) = dr/dt; v = ds/dt
- Instantaneous velocity (v) is the limit of the average velocity as the time interval approaches zero.
- v = lim (Δt→0) (Δr / Δt) = dr/dt, where dr is the infinitesimally small displacement in an infinitesimally small time dt.
- For very small intervals, the displacement (dr) is along the line of motion.
- The magnitude of the instantaneous velocity is the instantaneous speed. |dr|/dt = ds/dt
Average Acceleration
11
⚡ Quick Summary
Average acceleration is how much your velocity changes over a period of time. If you speed up or slow down, you are accelerating.
a_av = (v2 - v1) / (t2 - t1)
- Acceleration occurs when velocity changes with time.
- Average acceleration (a_av) is the change in velocity divided by the time interval.
- a_av = (v2 - v1) / (t2 - t1), where v1 and v2 are the velocities at times t1 and t2, respectively.
- Only the velocities at the start and end of the time interval matter for average acceleration.
Instantaneous Acceleration
11
⚡ Quick Summary
Instantaneous acceleration is how quickly your velocity is changing at a single moment. Think of it as the reading on an 'accelerometer' at one specific instant.
a = lim (Δt→0) (Δv / Δt) = dv/dt
- Instantaneous acceleration (a) is the limit of the average acceleration as the time interval approaches zero.
- a = lim (Δt→0) (Δv / Δt) = dv/dt, where dv is the infinitesimally small change in velocity in an infinitesimally small time dt.
- Instantaneous acceleration is also simply called 'acceleration'.
- The dimension of acceleration is LT^-2, and the SI unit is m/s^2.
Motion in a Straight Line
11
⚡ Quick Summary
When something moves along a straight line, we can describe its position with just one number (its x-coordinate) and track how that number changes over time.
N/A
- Motion is simplified when a particle moves on a straight line.
- The line is chosen as the X-axis, and a suitable time instant is taken as t=0.
- The origin is generally taken as the point where the particle is located at t=0.
- The position of the particle at time t is given by its coordinate x at that time.
Relative Position
Class 11
⚡ Quick Summary
Imagine you're in a car. Your position relative to a tree changes as the car moves. This section describes how to mathematically describe the position of something (a particle) from different viewpoints (frames of reference).
r_(P, S) = r_(P, S') + r_(S', S)
- Frame of Reference: A coordinate system from which observations and measurements are made.
- Position Vector: A vector that specifies the position of a point with respect to a chosen origin.
- If particle P has position vector `r_(P, S)` with respect to frame S, and position vector `r_(P, S')` with respect to frame S', and frame S' has position `r_(S', S)` with respect to frame S, then: `r_(P, S) = r_(P, S') + r_(S', S)`
Relative Velocity
Class 11
⚡ Quick Summary
Think about two cars moving. To someone standing still, each car has a speed. But to someone *in* one of the cars, the *other* car's speed is different (it's relative!). This section explains how to calculate those relative speeds.
v_(P, S) = v_(P, S') + v_(S', S)
v_(1, 2) = v_(1, S) - v_(2, S)
- The velocity of a particle with respect to a frame of reference.
- If particle P has velocity `v_(P, S)` with respect to frame S, and velocity `v_(P, S')` with respect to frame S', and frame S' has velocity `v_(S', S)` with respect to frame S, then: `v_(P, S) = v_(P, S') + v_(S', S)`
- The velocity of body 1 with respect to body 2 is obtained by subtracting the velocity of body 2 from the velocity of body 1. `v_(1, 2) = v_(1, S) - v_(2, S)`
Projectile Motion: Speed at an Angle
Class 11
⚡ Quick Summary
When a projectile is moving, its horizontal speed stays the same. So, if you know the initial speed and angle, and you want to find the speed at another angle, just use: v = u cos(θ) / cos(α). 'u' is starting speed, 'θ' is starting angle, and 'α' is the new angle.
v = u cos(θ) / cos(α)
- The horizontal component of a projectile's velocity remains constant throughout its motion (assuming no air resistance).
- Let 'u' be the initial speed of the projectile at an angle θ with the horizontal.
- Let 'v' be the speed of the projectile when its direction makes an angle α with the horizontal.
Projectile Motion: Hitting a Falling Object
Class 11
⚡ Quick Summary
If you fire a bullet horizontally at an object that starts falling the moment the bullet is fired, the bullet WILL hit the object. This is because both the bullet and the object fall vertically at the same rate due to gravity.
Vertical distance = 1/2 * g * t^2 (for both bullet and object)
- Consider an object starting to fall vertically at the same instant a bullet is fired horizontally towards it.
- Both the bullet and the object experience the same vertical acceleration due to gravity (g).
- The vertical distance traveled by both the bullet and the object in time 't' will be the same (1/2 * g * t^2).