Motion in Space
Position, velocity and acceleration
Suppose an object is moving in 3D space. We can describe its motion by specifying the position of the object at time . It can be expressed as a vector-valued function (the position vector of the object) for time .
The parametric curve defined by is the trajectory of the moving object.
The velocity of the object at time is . It measures the infinitesimal change of position of the object at time . The speed is the norm of velocity i.e. .
The acceleration of the object at time is , which is the infinitesimal change of velocity of the object at time .
A Projectile motion in 3D space
Suppose an object is launched from at the launch angle , where in the direction on the xy-plane (as shown in the applet below). Let be the initial speed of launching the object. We have
We assume the object is under the downward gravitational force. Then by Newton's second law of motion, , where is the gravitational acceleration on Earth.
Now we use integration to solve for and :
Let be the time when the object hits the ground again. Then we have
Let be the range of the projectile i.e. the distance from the origin (the launch location) to the place where the object hits the ground again. Then we have
Let be the maximum height of the trajectory. It occurs at . Therefore, we have
Exercise: Find the angle that has the maximum range.