A ball is thrown vertically upward with an initial velocity of 5 ft/sec from a height of 20 feet. Which function models its motion?
i literally am lost and have no idea how to this problem. if i could have help with the steps on how to do the problem and to get the correct answer that'd be great. I have a few more problems like this one and want to learn how to do them with good steps.
better check your text some more. With initial height h and initial velocity v, the height y at time t is
y(t) = h + vt - 16t^2
So, you have
y(t) = 20 + 5t - 16t^2
When working problems, don't forget what you know about parabolas -- their roots, vertex, etc.
To solve this problem, we need to understand the basic principles of motion and apply the appropriate equations.
Step 1: Understand the motion
The ball is thrown vertically upward, which means it moves in one dimension. The only forces acting on the ball are gravity pulling it down and the initial velocity pushing it upward.
Step 2: Identify the known and unknown values
Known values:
- Initial velocity (upward) = 5 ft/sec
- Initial height = 20 ft
Unknown:
- The function that models the motion
Step 3: Recall relevant equations
1. The equation for vertical motion is given by:
h(t) = −16t^2 + vt + h,
where h(t) represents the height of the object after t seconds, v is the initial velocity, and h is the initial height.
2. In this case, the initial velocity is upward, so we can substitute v = +5 ft/sec, and the initial height is 20 ft, so we can substitute h = 20 ft.
Step 4: Plug in the known values
Substituting the known values into the equation, we get:
h(t) = -16t^2 + 5t + 20
Therefore, the function that models the motion of the ball is:
h(t) = -16t^2 + 5t + 20
Step 5: Simplify (if necessary)
If required, you can simplify the equation further by performing any mathematical operations (such as multiplication, addition, or factoring) to make it more concise.
Remember, the equation h(t) represents the height of the ball at any given time t, considering only the forces of gravity and the initial velocity. This equation can be used to calculate the height at a specific time or solve other related problems.
I hope this explanation helps you understand how to solve this problem. If you have any further questions, please let me know!
To solve this problem, we can use the equation of motion for vertical motion under constant acceleration. Here are the steps to find the function that models the ball's motion:
Step 1: Identify the known values:
- Initial velocity (v0) = 5 ft/sec (upward)
- Initial height (h0) = 20 ft
- Acceleration due to gravity (g) = -32 ft/sec² (negative because it acts downward)
Step 2: Write the equation of motion:
The equation for vertical motion is given by:
h(t) = h0 + v0t + (1/2)gt²
Step 3: Substitute the known values into the equation:
h(t) = 20 + 5t + (1/2)(-32)t²
Step 4: Simplify the equation:
h(t) = 20 + 5t - 16t²
Step 5: Finalize the function:
The function that models the ball's motion is:
h(t) = -16t² + 5t + 20
This function gives the height of the ball as a function of time (t) after it is thrown.
If you have more problems like this, feel free to ask for step-by-step guidance on each one.