# Relationship between Launch height and range of a projectile

• Stanley_physics
In summary, an increase in launch height leads to a greater downward distance for the projectile to travel, which results in a longer air time and therefore a greater horizontal range. This can be explained by the principles of projectile motion, where an increase in height leads to an increase in air time and ultimately an increase in range. The equation S = (vcostheta(vsintheta + (v^2sin^2theta-2gh)^1/2)/-g, derived from the values obtained in our experiment, further supports this relationship between launch height and range."
Stanley_physics

## Homework Statement

In a practical that we conducted, we were investigating the relationship between Launch height and the range (horizontal distance it travels) of a projectile. We launched a projectile from various heights and measured the horizontal range it traveled before reaching the ground. We concluded that as height increases, range also increases. We are now trying to explain why an increase in launch means an increase in range.

2. The attempt at a solution
This is our attempt at explaining why this occurs.

"The results suggest that an increase in launch height causes an increase in range, as was predicted in the hypothesise. This relation can be linked directly to the physics concepts of projectile motion. Range depends upon two factors, horizontal velocity and air time of the projectile. Since velocity was constant throughout the investigation, the only factor that had an impact upon the range is air time. Increasing the height means that the projectile has a greater vertical distance to travel. The upwards motion of the projectile is unchanged regardless of the height, however the difference in range between each height occurs due to the increased or decreased downwards distance the projectile must travel which is the result of the launch height. Increasing the launch height increases the downward distance, giving the horizontal component of the velocity greater time to act upon the projectile and hence increasing the range. "

Also, we derived an equation for range (S) in terms of height (h).

S = (vcostheta(vsintheta + (v^2sin^2theta-2gh)^1/2)/-g

where v = initial velocity, theta = launch angle, and g=gravity

This is the equation in terms of the values that we got in our practical. v = 2.94 , theta = 45 degrees, g = -9.8

therefore , S = (2.94cos45)(2.94^2(sin45)^2-2(-9.8)(h))^1/2)/-(-9.8)

This maybe useful.

We were looking for a more scientific explanation which could maybe include mathematical explanations as well, however we could not create one ourselves. We couldn't find anything worthwhile on the internet either. It will be really useful if someone can help us out.

I think the explanation is fine. You can simplify it, however: Neglecting things like the air density changing with height (irrelevant at your parameters), increasing the launch height is equivalent to lowering the floor. The mass will always have the same horizontal distance when it crosses the height of the launch point again. Afterwards it continues to travel forwards, and the more height it has at this point (the lower the floor is relative to this point) the more it will travel forwards before hitting the ground.

Stanley_physics
I agree with the assessment by @mfb. However, I feel obliged to make a remark involving good habits that will serve you well in the future, especially when you draw and label force diagrams. Think of g as the magnitude of the acceleration of gravity, that is g = +9.8 m/s2. For projectile motion you start with the general equation for constant vertical acceleration y(t) = y0 + v0y t +(½)ay t2 and then adapt it tot he specific problem. Here, y0 = h, v0y t = v0sin(θ) and ay = -g. Therefore, y(t) = y0 + v0sin(θ) t - (½)g t2 where g is a positive quantity. Note how the component of the acceleration ay is given a negative algebraic sign indicating that the direction is "down". No more negative signs need to be introduced as you proceed with the algebra and, when the time comes to substitute numbers, you replace g with +9.8 m/s2.

Stanley_physics said:

## Homework Statement

In a practical that we conducted, we were investigating the relationship between Launch height and the range (horizontal distance it travels) of a projectile. We launched a projectile from various heights and measured the horizontal range it traveled before reaching the ground. We concluded that as height increases, range also increases. We are now trying to explain why an increase in launch means an increase in range.

2. The attempt at a solution
This is our attempt at explaining why this occurs.

"The results suggest that an increase in launch height causes an increase in range, as was predicted in the hypothesise. This relation can be linked directly to the physics concepts of projectile motion. Range depends upon two factors, horizontal velocity and air time of the projectile. Since velocity was constant throughout the investigation, the only factor that had an impact upon the range is air time. Increasing the height means that the projectile has a greater vertical distance to travel. The upwards motion of the projectile is unchanged regardless of the height, however the difference in range between each height occurs due to the increased or decreased downwards distance the projectile must travel which is the result of the launch height. Increasing the launch height increases the downward distance, giving the horizontal component of the velocity greater time to act upon the projectile and hence increasing the range. "

Also, we derived an equation for range (S) in terms of height (h).

S = (vcostheta(vsintheta + (v^2sin^2theta-2gh)^1/2)/-g

where v = initial velocity, theta = launch angle, and g=gravity

This is the equation in terms of the values that we got in our practical. v = 2.94 , theta = 45 degrees, g = -9.8

therefore , S = (2.94cos45)(2.94^2(sin45)^2-2(-9.8)(h))^1/2)/-(-9.8)

This maybe useful.

We were looking for a more scientific explanation which could maybe include mathematical explanations as well, however we could not create one ourselves. We couldn't find anything worthwhile on the internet either. It will be really useful if someone can help us out.
Hi!

Just wondering where you got your sources from. I'm trying to conduct the same experiment and I need some background citations.

Thanks!

rias2 said:
Hi!

Just wondering where you got your sources from. I'm trying to conduct the same experiment and I need some background citations.

Thanks!
The thread is nearly three years old, Stanley_physics is no longer a member, and I see no evidence that he used any sources.

rias2 said:
Hi!

Just wondering where you got your sources from. I'm trying to conduct the same experiment and I need some background citations.

Thanks!
Hi there!
I am also currently conducting the same experiment, and was wondering if you have gathered citations? If you could, will you be able to please link them? I am now in your shoes lol (you're not pressured !
Thanks

The thread is over three years old. The previous poster from two months ago - who wanted somebody else to do their homework prep for them - hasn't been around in a month. Is this a generational thing ? Please start your own thread, telling us what you have done, and where you're stuck.

juicebox_441 said:
Hi there!
I am also currently conducting the same experiment, and was wondering if you have gathered citations? If you could, will you be able to please link them? I am now in your shoes lol (you're not pressured !
Thanks

Hey! I actually did find some sources, although not very good ones. Unfortunately, I only have them in APA format, I'm sorry! 4.3 projectile motion – university Physics Volume 1. (2019). Retrieved February 12, 2021

Brown, R. (1992, September 01). Maximizing the range of a projectile. Retrieved February 12, 2021

Disabled World. (2020, October 07). Average height to Weight Chart: Babies to teenagers. Retrieved February 12, 2021

Henelsmith, N. (2016, May 12). Projectile Motion: Finding the Optimal Launch Height. Retrieved 2021.

Lucas, S. (2009). What affects the range of a projectile? Retrieved February 12, 2021

Srivastava, A. (2015, May). An Experiment On Projectile Motion. Retrieved 2021.

Vinopal, L. (2017, November 03). Harvard researchers Confirm optimal way to throw stuff. Retrieved February 18, 2021

hmmm27
hmmm27 said:
The thread is over three years old. The previous poster from two months ago - who wanted somebody else to do their homework prep for them - hasn't been around in a month. Is this a generational thing ? Please start your own thread, telling us what you have done, and where you're stuck.
Haha actually, I was desperately trying to find sources for my experiment after looking for about a week! Please don't assume things you couldn't possibly know about:)

rias2 said:
Haha actually, I was desperately trying to find sources for my experiment after looking for about a week! Please don't assume things you couldn't possibly know about:)
From the original post, the assignment consisted of trying to find out - by experiment - if there is a relation of launch height to range. Makes me wonder why you need citations in the first place, for bog standard kinematics equations.

Also, one of your cites :

Disabled World. (2020, October 07). Average height to Weight Chart: Babies to teenagers. Retrieved February 12, 2021

seems to be worth commenting on : spherical objects work best for (basic) projectile motion experiments.

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hmmm27 said:
From the original post, the assignment consisted of trying to find out - by experiment - if there is a relation of launch height to range. Makes me wonder why you need citations in the first place, for bog standard kinematics equations.

Also, one of your cites :

Disabled World. (2020, October 07). Average height to Weight Chart: Babies to teenagers. Retrieved February 12, 2021

seems to be worth commenting on : spherical objects work best for (basic) projectile motion experiments.

It’s actually for my sports exercise and health class! I was trying to find the average height for an adolescent, as I was measuring the correlation between human launch height and projectile range. I’m not sure how you conduct your experiments for school, but here, we do background research before we experiment:)

I still think you'd do better starting with cannonballs, before moving up to human test subjects : the aerodynamics is pretty different over long distances.

hmmm27 said:
I still think you'd do better starting with cannonballs, before moving up to human test subjects : the aerodynamics is pretty different over long distances.

Unfortunately, it is a sports health class so I have to test human subjects rather than tools.

Ah well, there's no accounting for taste, then. High school physics says the best angle is 45deg. Aerodrag says that's wrong and it's less than that. Aerodrag is right, but requires working knowledge of the drag properties of the object to know just how right.

Since your class isn't actually physics and presumably your looking for formulae/tables to integrate rather than practice, I'd again suggest a new thread ; either in the 'Physics' or 'Medical' subforums - depending on whether you need more help with kinematics or kinaesthetics - rather than dry-humping an old, only partially related thread.

[Edit: that'd be 'Physics : Classical Physics' subforum]

Last edited:
hmmm27 said:
Ah well, there's no accounting for taste, then. High school physics says the best angle is 45deg. Aerodrag says that's wrong and it's less than that. Aerodrag is right, but requires working knowledge of the drag properties of the object to know just how right.

Since your class isn't actually physics and presumably your looking for formulae/tables to integrate rather than practice, I'd again suggest a new thread ; either in the 'Physics' or 'Medical' subforums - depending on whether you need more help with kinematics or kinaesthetics - rather than dry-humping an old, only partially related thread. Ta

[Edit: that'd be 'Physics : Classical Physics' forum]

Nope! This thread was absolutely fine and definitely did help me. I was just responding, so I could help somebody else. Bye!

rias2 said:
Hey! I actually did find some sources, although not very good ones. Unfortunately, I only have them in APA format, I'm sorry!4.3 projectile motion – university Physics Volume 1. (2019). Retrieved February 12, 2021

Brown, R. (1992, September 01). Maximizing the range of a projectile. Retrieved February 12, 2021

Disabled World. (2020, October 07). Average height to Weight Chart: Babies to teenagers. Retrieved February 12, 2021

Henelsmith, N. (2016, May 12). Projectile Motion: Finding the Optimal Launch Height. Retrieved 2021.

Lucas, S. (2009). What affects the range of a projectile? Retrieved February 12, 2021

Srivastava, A. (2015, May). An Experiment On Projectile Motion. Retrieved 2021.

Vinopal, L. (2017, November 03). Harvard researchers Confirm optimal way to throw stuff. Retrieved February 18, 2021
Thank you soooo much! I also found some other sources, but my teacher required "a wide variety of sources" so I was wondering if you had any- and you did!
Have a great day and stay safe,
Cheers!

## 1. How does the launch height affect the range of a projectile?

The launch height has a direct impact on the range of a projectile. The higher the launch height, the longer the projectile will stay in the air and therefore, the greater the range will be. This is because the initial vertical velocity of the projectile will be higher, allowing it to travel further horizontally before hitting the ground.

## 2. Is there an optimal launch height for maximizing the range of a projectile?

Yes, there is an optimal launch height for maximizing the range of a projectile. This is known as the optimal angle of launch, which is typically around 45 degrees. This angle allows for the maximum horizontal distance to be covered while also accounting for air resistance and other factors.

## 3. How does air resistance affect the relationship between launch height and range of a projectile?

Air resistance plays a significant role in the relationship between launch height and range of a projectile. As the projectile travels through the air, it experiences resistance which reduces its speed and therefore, its range. Higher launch heights are more susceptible to air resistance, as the projectile spends more time in the air, leading to a shorter range.

## 4. Can the weight of the projectile affect the relationship between launch height and range?

Yes, the weight of the projectile can affect the relationship between launch height and range. Heavier projectiles will have a shorter range compared to lighter projectiles when launched at the same initial velocity and angle. This is because the heavier projectile will experience more air resistance, slowing it down and reducing its range.

## 5. Are there any other factors that can influence the relationship between launch height and range of a projectile?

Yes, there are other factors that can influence the relationship between launch height and range of a projectile. These include the initial velocity, angle of launch, air density, and wind speed. All of these factors can impact the flight of the projectile and therefore, affect its range.

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