# Projectile motion: find distance a ball would land

• phys1213
In summary: Having t you can find d as a function of v0 (from the horizontal motion equation). The rest is algebra.In summary, the conversation discusses the process of determining an equation to find the distance a ball will travel from a spring loaded projectile launcher. The relevant variables include the spring constant, compression, mass of the ball, height of the launcher, and angle it is set at. The conservation of energy equation is mentioned, as well as other equations such as F=ma and KE. The attempt at a solution involves using the conservation of energy equation and integrating velocity with respect to time to find the distance. However, it is mentioned that a projectile motion equation is needed to complete the solution.

## Homework Statement

I'm trying to come up with an equation to determine where a ball would land (basically the distance it moves) from a spring loaded projectile launcher set up on a table. I'm looking for "d", and I know the spring constant, compression, mass of the ball, height the ball starts at, angle the launcher is set at, and whatever else I can measure using a meter stick, balance, and protractor. There aren't any numbers just known variables

## Homework Equations

Conservation of Energy eqn (at least the version I learned in class): Efinal-Eintial=Einput-Eoutput
F=ma
Fg=mg
sinθ=voy/vo
cosθ=vox/vo
KE: 1/2mv2
Spring: 1/2kx2

## The Attempt at a Solution

I attempted to use conservation of energy but I get stuck trying to figure out where d goes into be able to solve for it. Also, other online resources use a conservation of energy eqn that has different terms than what I was taught, but I'm assuming they are all the same.
System: Ball and Earth
Initial time: just after ball leaves launcher
final time: just before ball hit ground
Efinal=1/2mvf2
Einitial= 1/2mvo2+1/2kx2+mgH
Einput-Eoutput=0
1/2mvf2-1/2mvo2-1/2kx2-mgH=0
And then I'm stuck trying to figure out how the distance goes into this. I'm wondering whether I need to integrate the velocity with respect to time and relate that to the distance since the distance the ball travels is the velocity*time. Any help is appreciated!

Hi phys,

Not bad for a first post. Yes, you need some projectile motion equation to complete this. From initial Einitial= 1/2mvo2+1/2kx2+mgH (where vo = 0 ?) you get v0. And your projectile trajectory is uniform motion horizontally (needing t) and uniformly accelerated vertically (which gives you a quadratic equation for t) .

## 1. How is the distance a ball would land calculated in projectile motion?

The distance a ball would land in projectile motion is calculated using the formula d = v0t + 1/2at2, where d is the distance, v0 is the initial velocity, t is the time, and a is the acceleration due to gravity.

## 2. Does the mass of the ball affect the distance it would land in projectile motion?

No, the mass of the ball does not affect the distance it would land in projectile motion. The distance is determined by the initial velocity and the angle at which the ball is launched.

## 3. How does the angle of launch affect the distance a ball would land in projectile motion?

The angle of launch affects the distance a ball would land in projectile motion because it determines the vertical and horizontal components of the initial velocity. A higher angle of launch will result in a longer horizontal distance, while a lower angle of launch will result in a shorter horizontal distance.

## 4. Can air resistance affect the distance a ball would land in projectile motion?

Yes, air resistance can affect the distance a ball would land in projectile motion. This is because it acts in the opposite direction of the ball's motion and can slow it down, reducing its range. However, for most practical scenarios, the effects of air resistance are negligible.

## 5. How can the distance a ball would land in projectile motion be increased?

The distance a ball would land in projectile motion can be increased by increasing the initial velocity, increasing the angle of launch, or reducing the effects of air resistance. Additionally, launching the ball from a higher elevation or using a more aerodynamic ball can also increase the distance it would land.