Physics 141MechanicsLecture 4 Motion in 3-D Motion in 2-dimensions or 3-dimensions has to be described by vectors. However, what we have learnt from 1-dimensional kinematics is of great value since we can treat each dimension separately as a 1-D problem. The motion of a particle is described by the position vector r=r(t)=x(t)i+y(t)j+z(t)k, a vector pointing from the origin to (x,y,z), the coordinates of the particle. The components x(t), y(t), z(t) are functions of time t and can have forms very different from each other. The displacement of the particle is described by the displacement vector  r=r(t 2 )-r(t 1 )=(x(t 2 )-x(t 1 ))i+(y(t 2 )-y(t 1 ))j+(z(t 2 )-z(t 1 ))k. Note here the unit vectors are not changing with time.

Velocity in Higher Dimensions The average velocity is again defined as the displacement divided by the time interval taken for the displacement The instantaneous velocity, or velocity in short, is the time derivative of the position vector Speed

Acceleration in Higher Dimensions The average acceleration is defined as the velocity change divided by the time interval taken for the velocity change The instantaneous acceleration, or acceleration in short, is the time derivative of the velocity vector Magnitude

Motion of Constant Acceleration If the acceleration a is a constant vector, then Note that both the magnitude and direction are constant. Similar to the 1-D case, we have then Each of the above equations are in fact three equations

Projectile Motion One common and useful motion of constant acceleration in higher dimensions is that of a projectile. We have discussed projectile motion confined in 1-D: the free fall body. What if the initial velocity is not vertical? Let’s formulate the motion of projectile. If we ignore the air friction, the only acceleration is due to the gravity in the vertical direction. The projectile will move within the plane defiled by the initial velocity and the vertical direction. Let’s set up the coordinate system with XOY in this plane. There is no motion in the third direction and we don’t have to worry about it. We further define x to be horizontal and y vertical.

Assume the initial velocity of the projectile is v 0 at an angle  to the horizontal. We choose the coordinate system shown on the right: The initial velocity is v 0 = v 0 cos  i + v 0 sin  j and the constant acceleration is a=- gj. The initial position is the origin. The motion along horizontal direction is x(t) = v 0 t cos  In the vertical direction, The maximum height is reached when v y (t max ) = 0, or  x v0v0 g y 0

The maximum height To get the range of the projectile, we just have to let y(t)=0

Example: A basket ball is thrown with an angle  at the basket distance D away and height H above the ground. What should be the speed v 0 of the ball? Solution: This is a projectile problem with the final y position at H. Eliminate t we get

Demonstration: Monkey and Cannon Suppose you are in a jungle with a huge cannon. You see a monkey and decided to shoot it with your cannon. At the moment you fires, the monkey falls down. How would you have to aim to get a hit, disregard the moral issues and air friction. It seems a complicated problem since you have to consider a moving target. In reality, all you have to do is just to aim straight at the monkey. If there were no gravity, it’s obviously right. With the gravity turned on, though, both the monkey and the cannon ball fall the same amount in the same time if you aimed right.

Uniform Circular Motion If the position of a particle is its trajectory is a circle with radius R. The angle of the position vector with the x-axis,  (t)=  t, increases linearly with time.  he angular velocity described the rate of angular change. In this case, Such motion is called uniform circular motion. The period of uniform circular motion is the time taken for one full revolution, or  and it is T=2  x y r tt

Velocity of Uniform Circular Motion The velocity of a uniform circular motion is The magnitude of the velocity is constant Also v is always perpendicular to r since

Acceleration of Uniform Circular Motion The acceleration of a uniform circular motion is The magnitude of the acceleration is also constant Also a is always perpendicular to v since And a is always antiparallel to r, pointing to the center of the motion. It is therefore named centripetal acceleration.