Let’s suppose you throw a ball straight up with an initial speed of 50 feet per second from a height of 6 feet.

a.Find the parametric equations that describe the motion of the ball as a function of time.
b.How long is the ball in the air?
c.Determine when the ball is at maximum height. Find its maximum height.

i haven't got the slightest idea of how to do this.

a = -32 ft/s^2

v = 50 -32 t
h = 6 + 50 t - 16 t^2

find t when h = 0 (the positive answer)

max height is when v = 0, find t and h then.

To solve this problem, we can use the equations of motion for a projectile in free fall. There are a few key equations that we need to keep in mind:

1. The position of the ball in the vertical direction (y) can be described by the equation:
y = y0 + v0t - (1/2)gt^2,
where y is the position of the ball at time t, y0 is the initial height, v0 is the initial velocity, g is the acceleration due to gravity (approximately 32.2 ft/s^2), and t is time.

2. The velocity of the ball in the vertical direction can be described by the equation:
vy = v0 - gt,
where vy is the velocity of the ball at time t.

Now, let's solve the problem step by step:

a. Find the parametric equations that describe the motion of the ball as a function of time:

We want to find equations for both the vertical position and the vertical velocity of the ball as functions of time.

Using equation (1), we have:
y = 6 + 50t - (1/2)(32.2)t^2,

Using equation (2), we have:
vy = 50 - 32.2t.

b. To find how long the ball is in the air, we need to determine when the ball hits the ground. At that point, the vertical position will be y = 0. We can set up the equation:

0 = 6 + 50t - (1/2)(32.2)t^2.

Simplifying and rearranging the equation, we get a quadratic equation:
16.1t^2 - 50t - 6 = 0.

Solve this equation to find the value(s) of t when the ball hits the ground. In this case, since the initial velocity is upward, it will take the ball some time to reach the ground.

c. To determine when the ball reaches the maximum height, we need to find the time when the ball's velocity in the vertical direction becomes zero. At this point, the ball will start falling back down after reaching its maximum height.

Set vy = 0 in equation (2):
0 = 50 - 32.2t.

Solve this equation to find the time when the ball reaches its maximum height. Then, substitute this time back into equation (1) to find the corresponding maximum height.

Using these steps, you should be able to solve the problem and find the answers to parts (b) and (c).