A written summary for the equation

S = -16t^2+v*t+k. Need help explaining the results. Thanks. It is the scenario where a ball is thrown into the air problem.

What is S at time zero? What does S do as time progresses? Does S have a maximum? When is S equal to the initial position?

The v term is the initial upward velocity, and the -16 is the acceleration of gravity, in ft/s^2 units

Sure! Let's break down the given equation and understand its meaning in the context of a ball being thrown into the air.

The equation S = -16t^2 + vt + k represents the height of the ball, S, as a function of time, t, when it is thrown into the air. The variable v represents the initial velocity of the ball, and k represents any additional height or displacement before the ball is thrown.

In this equation, the term -16t^2 represents the effect of gravity on the height of the ball. The negative sign indicates that the ball is moving in the opposite direction of gravity (upwards) when t is increasing. The term 16t^2 indicates that the ball's height decreases quadratically with time due to the acceleration caused by gravity.

The term vt represents the impact of the initial velocity of the ball. If the ball is thrown upwards, v is positive, and the height of the ball will increase. If it is thrown downwards, v is negative, and the height will decrease. The magnitude of v affects the steepness of the height curve.

Lastly, the constant k represents any additional height or displacement before the ball is thrown. This could include an initial height or any height gained or lost before the ball starts its upward or downward trajectory. Adding or subtracting k shifts the entire height curve up or down accordingly.

To fully understand the output of this equation, you would need to provide specific values for the variables v and k, as well as the time interval (t). This would allow you to calculate the height of the ball at any given point in time.

Additionally, it's worth noting that if you're looking for specific statistics like the maximum height reached or the time it takes for the ball to hit the ground, you would need to manipulate or analyze the equation further, such as finding the vertex of the quadratic function or solving for the time when the height is zero.

I hope this explanation helps clarify the equation and how it relates to the scenario of a ball being thrown into the air!