Physics
Motion in a Plane
Projectile Motion - Definitions
Class 11
⚡ Quick Summary
Imagine throwing a ball! Projectile motion describes its path. Key terms: Point of projection (where you throw it from), angle of projection (how high you aim), range (how far it goes), and time of flight (how long it's in the air).
- Point of Projection: The point from which the particle is projected (O in Figure 3.10).
- Angle of Projection (θ): The angle between the initial velocity vector and the horizontal.
- Horizontal Range (R): The horizontal distance traveled by the projectile before hitting the ground (OB in Figure 3.10).
- Time of Flight (T): The total time for which the projectile remains in the air.
Projectile Motion - Horizontal Motion
Class 11
⚡ Quick Summary
Horizontally, the projectile moves at a constant speed because there's no acceleration in that direction (we're ignoring air resistance).
v_x = u cosθ; x = ut cosθ
- Acceleration in the horizontal direction (ax) is zero.
- Horizontal component of velocity (vx) remains constant: vx = ux = u cosθ.
- Horizontal displacement (x) at time t is given by: x = uxt = ut cosθ.
Projectile Motion - Vertical Motion
Class 11
⚡ Quick Summary
Vertically, gravity pulls the projectile down, causing it to slow down as it goes up and speed up as it comes down. This is like throwing a ball straight up in the air.
v_y = u sinθ - gt; y = (u sinθ)t - (1/2)gt^2; v_y^2 = (u sinθ)^2 - 2gy
- Acceleration in the vertical direction (ay) is -g (acceleration due to gravity, acting downwards).
- Vertical component of initial velocity is uy.
- Vertical component of velocity (vy) at time t: vy = uy - gt = u sinθ - gt.
- Vertical displacement (y) at time t: y = uyt - (1/2)gt2 = (u sinθ)t - (1/2)gt2.
- Relationship between final vertical velocity, initial vertical velocity, and vertical displacement: vy2 = uy2 - 2gy = (u sinθ)2 - 2gy.
Meeting Point of Particles Moving Towards Each Other
11
⚡ Quick Summary
When multiple particles are moving towards each other in a symmetrical pattern (like an equilateral triangle), they will eventually meet at the center (centroid) of the pattern. You can find the time it takes for them to meet by focusing on one particle and calculating how long it takes for it to reach the center.
Time = d/(v*cos(θ)), where d is the initial distance from the particle to the meeting point, v is the constant speed, and θ is the angle between the particle's velocity and the line connecting the particle to the meeting point.
When particles move towards each other with constant speed 'v', forming a symmetrical shape (e.g., an equilateral triangle), they meet at the centroid (O) of the shape.
To find the time to meet:
1. Consider one particle, say A.
2. Find the component of its velocity along the line connecting it to the centroid (AO). This component represents how quickly the distance AO is decreasing.
3. Calculate the initial distance AO.
4. Time to meet = (Initial distance AO) / (Velocity component along AO)
Alternative Method to find Meeting time
11
⚡ Quick Summary
Instead of focusing on the distance to the center, look at how quickly the distance between any two particles is decreasing. Calculate the rate at which they're approaching each other, and then divide the initial distance by that rate to find the time to collision.
t = d / (v + v*cos(θ)), where d is the initial distance between two particles, v is their speed, and θ is the angle between their velocities
1. Consider two adjacent particles A and B. Find the component of B's velocity that is directed *towards* A.
2. Calculate the *relative* speed at which A and B are approaching each other. This is the sum of A's speed along the line joining them and the component of B's velocity towards A.
3. Time to meet = (Initial separation between A and B) / (Relative speed).
Circular Motion
11
⚡ Quick Summary
The motion of an object along a circular path.
Centripetal Acceleration: a = v^2/r; Centripetal Force: F = mv^2/r
Explains centripetal acceleration and force, uniform and non-uniform circular motion.