# 12.3 Velocity and Acceleration

### 12.3

Velocity and Acceleration

### 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