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2D Motion. Oct. 2013. Contents. Displacement, Velocity and Acceleration Equation of Kinematics in 2D Projectile motion. Displacement, Velocity and Acceleration. 2D motion refers to situations in which the motion is along a curved path that lies in a plane
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2D Motion Oct. 2013
Contents • Displacement, Velocity and Acceleration • Equation of Kinematics in 2D • Projectile motion
Displacement, Velocity and Acceleration 2D motion refers to situations in which the motion is along a curved path that lies in a plane Such motion can be described using the same concepts as used for 1D motion Due to the curved path, the location vector r at each point in time is pointing in different direction (and not parallel to line as in 1D)
Terms - Displacement The race car is at two different positions along the curve, r and r0 The displacement of the car is the vector drawn from the initial position r0 at time t0 to the final position r at time t The magnitude of r is the shortest distance between the two positions
Terms – Average Velocity The average velocity of the car between two positions is defined in a manner similar to 1D motion The average velocity vector has the same direction as the displacement vector The average velocity v becomes equal to the instantaneous velocity in the limit that t becomes infinitesimally small ( t 0 s)
Terms – Average acceleration V is tangent to the path of the car. The vector components Vx and Vy of the velocity, are parallel to the x and y axes The average acceleration is defined just as it is for 1D motion, as the change in velocity, divided by the elapsed time
Terms The average acceleration has the same direction as the change in velocity In the limit that the elapsed time becomes infinitesimally small, the average acceleration becomes equal to the instantaneous acceleration The acceleration has a vector component ax along the x direction and a vector component ay along the y direction
Equation of Kinematics in 2D Since displacement, velocity and acceleration are vectors, each have 2 components in 2D. e.g:
Equation of Kinematics in 2D – cont. A spacecraft equipped with two engines that are mounted perpendicular to each other Engine parallel to X
Equation of Kinematics in 2D – cont. It is important to realize that: the x part of the motion occurs exactly as it would if the y part did not occur at all. Similarly, the y part of the motion occurs exactly as it would if the x part of the motion did not exist In other words, the x and y motions are independent of each other. Engine parallel to Y
Example 1 In the +x direction, the spacecraft has:an initial velocity component of v0x= +22 m/s an acceleration component of ax= +24 m/s2. In the y direction, the analogous quantities are v0y= +14 m/s ay= +12 m/s2 At a time of t=7.0 s, find the x and y components of the spacecraft’s displacement?
Check Equation Solution 1 The motion in the x direction and the motion in the y direction can be treated as a 1D motion subject to the equations of kinematics for constant acceleration
Example 1 – cont. Find the spacecraft’s final velocity (magnitude and direction)?
Check Equation Solution 1 – cont. = =
Projectile motion Projectile motion is a 2D motion of a free body – the only impact on the motion of free body is the acceleration due to gravity in downward direction.
Projectile motion has initial velocity Vo directed α above the horizon V0 α
Superposition principal We consider the horizontal and vertical parts of the motion separately In the horizontal or x direction, the moving object (the projectile) does not slow down in the absence of air resistance. The x component of the velocity remains constant at its initial value or vx=v0x The x component of the acceleration is ax=0 m/s2
The monkey release the branch when the banana is fired. At what direction should the banana be fired so that the monkey will catch it?
Both the banana and the monkey feel the gravity force, and their movement in Y direction is impacted the same
The speed of the banana will impact only the meeting point with the monkey
No forces act on the horizontal axis, hence this is constant velocity motion The velocity X component , maintains its magnitude and direction all the way The velocity Y component , changes its magnitude: starts by slowing down till zero, and than change direction and start to speed up The gravity works in the vertical direction downward, hence we have constant accelerated motion with a rate of 9.8 [m/s2]
At t=0 the body has a velocity Vo directed α above the positive X direction. We can calculate the componenets: y V0 V0y α x V0x
The velocity Y component , changes its magnitude because of gravity force: starts by slowing down till zero, and than change direction and start to speed up y V0 V0y α x V0x
At each point during the flight we can calculate the velocity by using: Note the negative g due to its direction y
y V Vy θ x Vx To calculate the resultant velocity magnitude we use: :It’s direction will be α is the angle between the initial velocity direction and the horizon θ is the angle between the velocity direction at some point and the horizon
Can you draw the graph of the horizontal velocity component? vx V0cosα t
vy Can you draw the graph of the vertical velocity component? V0sinα t
v Can you draw the graph of the resultant velocity magnitude? V0 V0cosα t
A package dropped from a plane flying in constant speed to the right. Attention!!! The package moves horizontally parallel to the plane since it has an initial speed equal to the plane speed. Relative to the plane, the package is falling straight down
Ball thrown straight up from a truck moving in constant speed. Relative to the truck, the ball is moving straight up
Body location equations: (start location at the origin, positive is up) X location from the origin Y location from the origin
Summary Topics covered: • Displacement, Velocity and Acceleration • Equation of Kinematics in 2D • Projectile motion Next meeting: Forces and Newton’s laws
