### 12.3

Velocity and Acceleration

### Projectile Motion

### Examples

A baseball player at second base throws a ball 90 feet to the player at first base. The ball is thrown

at 50 miles per hour at an angle of 15 above the horizontal. At what height does the player at first base catch the ball if the ball is thrown from a height of 5 feet?

A projectile is fired from ground level at an angle of 12

with the horizontal. Find the minimum initial velocity if the projectile is to have a range of 150 feet.

### 12.4

Tangent Vectors and Normal

Vectors

### Examples:

1) Find the unit tangent vector and find a set of parametric equations for the line tangent to the space curve at the point P.

r ( t )

t , t , 4

* t*

2

P ( 1 , 1 , 3 )

2) Find the principal unit normal vector to the curve above at the specified point.

### Sometimes shortcut to principal unit normal vector.

*For plane curves, you can find the principal unit normal vector by finding T (t)= x ( t ) i*

y ( t ) j and then observe that N (t) will either be

*N*

2

( t )

* y ( t ) i*

* x ( t ) j OR N*

2

( t )

* y ( t ) i*

* x ( t ) j*

It will be the one that points toward the concave side of the curve.

### Examples:

*1) Find T (t), N a for the given curve r(t) a*

*N r t*

*2 i*

*2 r e t cos ti*

* e t sin tj*

* e t*