Hi,

I have a math problem that I can't understand how to write the equation for. The question says: a ball is thrown upwards from an initial height of 20 meters at an initial velocity of 10 meters per second. Write an equation for the height (h) of the ball at time (t). My teacher gave us this equation, y= -gt + vo t (for time) +ho. That doesn't make sense with the problem. I'm not suer what format I would put the equation in. Can someone please help me solve this?

ho=20m

vo=10m/s
g=9.8m/s^2
Use the equation.

Thank you.

Of course, I'll be happy to help you understand how to write the equation for this problem.

In this scenario, the ball is thrown upwards, so we can assume it is subject to the force of gravity. The equation your teacher provided, y = -gt + vo t + ho, is a general equation for the position (y) of an object under the influence of gravity with an initial velocity (vo) and initial position (ho) at time (t). However, it seems there might be a mistake in your teacher's equation with the variable names.

To write the equation for the height (h) of the ball at time (t), we can start by identifying the variables given in the question:

Initial height (ho) = 20 meters
Initial velocity (vo) = 10 meters per second
Gravity (g) = 9.8 meters per second squared (assuming the acceleration due to gravity is 9.8 m/s^2)

The height (h) of the ball at any time (t) can be represented by the equation:

h = ho + vo t + (1/2)gt^2

Let me explain how this equation was derived:

To determine the height (h), we add the initial position (ho) to the displacement caused by the initial velocity (vo) over time (t).

First, consider the vertical motion of the ball due to its initial velocity (vo) only. The distance covered vertically by the ball due to its initial velocity over time can be represented by the equation:

Distance = Initial Velocity × Time = vo × t

Next, we need to consider the effects of gravity on the ball while in motion. The height (h) due to gravity can be represented by the equation:

Height due to gravity (h_g) = (1/2) × g × t^2

Where (1/2) is the constant factor resulting from the integration of the acceleration due to gravity with respect to time.

To find the total height (h) at any time (t), we need to sum the initial height (ho), the distance due to the initial velocity (vo × t), and the height due to gravity (h_g). This gives us the equation:

h = ho + vo t + h_g

Substituting (1/2) × g × t^2 for h_g, we get:

h = ho + vo t + (1/2)gt^2

Therefore, the equation for the height of the ball at time (t) is:

h = ho + vo t + (1/2)gt^2

In this specific problem, with the given values of ho = 20 meters, vo = 10 meters per second, and g = 9.8 meters per second squared, you can substitute these values into the equation to get the specific equation for this scenario:

h = 20 + 10t - 4.9t^2

I hope this explanation clarifies how to write the equation for the height (h) of the ball at time (t).