How To Find Maximum Height Physics

How To Find The Maximum Height Of A Projectile

How to Solve for Maximum Height in Physics : Fun With Physics

Maximum height of the object is the highest vertical position along its trajectory. The object is flying upwards before reaching the highest point – and it’s falling after that point. It means that at the highest point of projectile motion, the vertical velocity is equal to 0 .

0 = Vy g * t = V * sin g * th

From that equation we can find the time th needed to reach the maximum height hmax:

th = V * sin / g

The formula describing vertical distance is:

y = Vy * t g * t² / 2

So, given y = hmax and t = th, we can join those two equations together:

hmax = Vy * th g * th² / 2

hmax = V² * sin² / g g * / g)² / 2

hmax = V² * sin² /

And what if we launch a projectile from some initial height h? No worries! Apparently, the calculations are a piece of cake – all you need to do is add this initial elevation!

hmax = h + V² * sin² /

Let’s discuss some special cases with changing angle of launch:

Why An Object Thrown Upwards Comes Down After Reaching A Point

When an object is thrown with certain initial velocity , it gains a Kinetic energy at that moment of throwing. As it moves upwards from its initial position and gains height, its potential energy rises. This happens because Potential Energy is directly proportional to the height of the object. . Now from the Law of conversation of energy, we can say this rise in PE is happening at the cost of some form of energy being transformed. Here its the kinetic energy of the object which is expressed as 0.5 m V^2. As height rises, velocity falls which results in a reduction of KE and a corresponding rise in PE. At one point KE becomes zero. At that point, velocity becomes zero. This is called the highest point for an upward vertical movement. After this, the ball starts falling downwards.

Differently, we can say that the KE availed by the thrown object gets corroded under the negative influence of oppositely directing gravity . The influence is negative because gravity is pulling downwards while the ball is trying to move upwards. After some time when the entire KE gets nullified, the ball stops. And then starts falling towards the earths surface.

In all the above discussions, we have considered Air Resistance as negligible.**Those who are aware of escape velocity, you can read a post on it here: Escape Velocity

How Do You Find The Maximum Height Of A Projectile Calculus

4.6/5

Then, how do you find the maximum height of a projectile?

if = 90°, then the formula simplifies to: hmax = h + V0² / and the time of flight is the longest.

if = 45°, then the equation may be written as:

if = 0°, then vertical velocity is equal to 0 , and that’s the case of horizontal projectile motion.

One may also ask, what is maximum height of a projectile? Maximum Height Formula. The maximum height of the object is the highest vertical position along its trajectory. The maximum height of the projectile depends on the initial velocity v0, the launch angle , and the acceleration due to gravity. The unit of maximum height is meters .

Likewise, people ask, how do you find the height of a function?

Projectile Motion. An object is thrown straight up from the top of a building h feet tall with an initial velocity of v feet per second. The height of the object as a function of time can be modeled by the function h = 16t2 + vt + h, where h is the height of the object t seconds after it is thrown.

What is the maximum height the ball reaches?

The required maximum height of the ball is 208 feet.

Don’t Miss: How To Score 100 In Chemistry Class 12

How To Calculate Maximum Height Of A Projectile

The following are the simple and easy steps to evaluate projectile maximum height. Go through these steps and follow them carefully.

Get the initial velocity of the object.
the angle of launch and initial height of the object from the given question.
Square the initial velocity, find the sine of angle square and multiply them.
Double the acceleration due to gravity which is 9.81.
Divide the result obtained in step 3 by step 4.
Add the result to the initial height.
The obtained result is called the maximum height of the projectile motion.

Derive The Equation Of The Time Taken By The Ball To Reach The Maximum Height During Its Upward Movement

Projectile Motion: Finding the Maximum Height and the ...

As this acceleration due to gravity is working opposite to the upward velocity we have to use a negative sign in the formula below, used for the upward movement of the ball. We know the value of g in SI is 9.8 m/second square.

Using one of the equations of motion,

v2 = v1 gt

As v2=0, , then we can write the previous equation as follows:

0 = v1 gt

or, t = v1/g .

So from equation , thetime taken by the ball to reach the maximum height is expressed as = / ..

So just for example, if a ball is thrown vertically upwards with 98 m/s velocity, then to reach the maximum height it will take = 98/9.8 =10 seconds.

Calculate The Range Of The Projectile:

The range of the projectile is calculated by the projectile motion solver, where the total horizontal distance is the distance traveled during the flight time. Again, if were throwing the object from the surface where the initial height is zero, then we can write the formula as:

R = t * V_x = V_y * 2 * V_x / g

It may changed into the form of:

R = sin * V_2 / g

How Do You Measure The Height Of A Rocket

HeightrocketfindrocketrocketThe calculator uses the standard formula from Newtonian physics to figure out how long before the falling object goes splat:

The force of gravity, g = 9.8 m/s2

Time to splat: sqrt

Velocity at splat time: sqrt

Energy at splat time: 1/2 * mass * velocity2 = mass * g * height.

Don’t Miss: Linear Algebra Gilbert Strang Notes

How To Find Maximum Height Of A Projectile

Ever wanted to know how high and how fast you can kick a football? Turns out, its pretty easy to figure out. All you need to know is how far away from you the ball lands and how much time it spends in the air . Combine these two simple measurements with a little bit of physics and you can start competing with your friends to see who can kick the highest or the fastest!

Find Out The Formula Of The Time Period For The Downward Movement When A Ball Is Thrown Vertically Upward

Projectile Motion Calculating the Maximum Height Part 6

It can be proved that the time for the downward movement or the time taken by the ball to fall from the highest point and reach the ground is same as the time required for the upward movement = v1/g

Lets prove it here mathematically:

v3 = v2 + g Tor, v3 = 0 + g T

so, T = v3/g = v1/g .

So Time for downward movement = Time for upward movement = v1/g

That means, time for downward travel = time of upward travel

Total time required for the upward and downward movement

So for a vertically thrown object the total time taken for its upward and downward movements = t + T=2v1/g .

Don’t Miss: Practice 2 4 Reasoning In Algebra Answer Key

Faq’s On Projectile Motion Maximum Height Calculator

1. How to get maximum height in projectile motion?

The simple formula to calculate the projectile motion maximum height is h + Vo/sub> ² * sin² / . Students have to obtain the angle of launch, initial velocity, initial height and substitute those in the given formula. Evaluate the expression to get the maximum height of the projectile motion.

2. Does height matter in projectile motion?

Yes, the height of the object affects the trajectory of the projectile. For any speed of release, angle of release, the horizontal displacement increases as the height of release increases.

3. How do you find the projectile maximum height on a calculator?

Students have to enter data in the required input fields of the Maximum height calculator -projectile motion such as initial height, angle of launch, initial velocity. Hit on the calculate button to get the maximum height as output in the fraction of seconds.

4. What is projectile motion?

Projectile means an object which is flight after thrown or projected into the air. In projectile motion, the only acceleration acting is in the vertical direction and that is the acceleration due to gravity. Examples are football, baseball, cricket ball and so on.

Favorite Calculators

How To Calculate The Maximum Height Of A Projectile

How to calculate maximum projectile height

First, determine the initial velocity.

This is the total velocity of the object.

Next, determine the angle of launch

This is the angle measured with respect to the x-axis.

Calculate the maximum height.

Enter the total velocity and angle of launch into the formula h = V² * sin² / to calculate the maximum height.

You May Like: Geometry Dash Eric Van Wilderman

Projectile Range Calculator Example

Let’s find out the range of the hypothetical volcanic ballistic projectiles :

Enter the projectile velocity. Assume that a rock was ejected from the volcano with a speed of 30 m/s.

Type in the angle. Let’s say that the angle was equal to 25°.

Fill in the box of initial height. Our volcano was quite small and the rock lands on the side of the volcano at 100 m lower altitude than the launching point.

The projectile range calculator shows the answer! The maximal distance is 162.87 m for our volcano example. That’s a lot!

In reality, the projectile motion is a much complicated phenomenon. The air resistance reduces a range of a projectile – and that’s dependent on the object volume, shape, surface smoothness, mass, etc.

What Is Projectile Motion

In physics, a projectile motion is defined as the motion of an object thrown into the air and subjected to gravitational acceleration. The path followed by the object is called the trajectory, while an object is indicted as the projectile, and the movement of the object is called the motion of the projectile.

The projectile motion of the object can be in any form, such as straight path, circular trajectory, parabola, hyperbola, ellipse, etc. The projectile motion equations are used in engineering structures. The motion of the projectile provides a clear idea of the motion of an object that is close to the earths surface and accelerated by the gravity near the surface of the earth.

Also Check: What Is Amu In Chemistry

What Is The Acceleration Of A Ball Thrown Vertically Upwards During Upward Movement

The acceleration of the ball would be equal to the acceleration due to gravity caused by gravitational pull or force exerted by the earth on the ball. Its value is approximately 9.8 m/s^2 and its direction would be downwards towards the center of the earth.

Its pretty evident that after the upward throw, the velocity of the ball gradually decreases i.e. a negative acceleration was working on the ball. The acceleration is negative because this acceleration is directing downwards while the ball is moving upward.And because of this negative acceleration, the velocity of the ball is gradually decreasing. And Yes. As said above, this acceleration is nothing but the acceleration due to gravity caused by gravitational pull or force exerted by the earth on the ball. Its value is generally taken as 9.8 m/s^2. See here how acceleration due to gravity varies with height and depthwrt the surface of the earth.

How To Find Equation Of Path Of Projectile

Let a Particle be projected with velocity v0 and after time t it acquire a position P

x = v0cos0 t

$ \displaystyle t = \frac $

$ \displaystyle y = v_ t \fracg t^2$

$ \displaystyle y = v_0 sin\theta_0 t \fracg t^2 $

$ \displaystyle y = v_0 sin\theta_0 \fracg ^2 $

$ \displaystyle y = x tan\theta_0 \frac$

This is the equation of a parabola. Therefore the path of the particle is parabolic when the particle passes the level of projection.

Don’t Miss: How Does Quantum Physics Explain Spirituality

The Formula For Maximum Height

The maximum height of the object in projectile motion depends on the initial velocity, the launch angle and the acceleration due to gravity. Its unit of measurement is meters. So Maximum Height Formula is:

Mathematically: \

H is maximum height
g is acceleration due to gravity, i.e. \\)
\ is the angle of the initial velocity from the horizontal plane

What Happens When A Ball Is Thrown Vertically Upward

Physics 3: Motion in 2-D Projectile Motion (29 of 31) Find Maximum Height=? (General Equation)

When a ball is thrown vertically upward it starts its vertical motion with an initial velocity. As it moves upwards vertically its velocity reduces gradually under the influence of earths gravity working towards the opposite direction of the balls motion. In other words, during upward movement, the ball is moving with retardation. And finally, the velocity of the ball becomes zero at a height. Then again it starts falling downwards vertically and this time its velocity increases gradually under the influence of gravity. During downward movement balls direction is the same as that of gravity and as a result, the ball comes down with acceleration and reaches the ground.

Don’t Miss: Algebra De Baldor Libro Completo Online

Maximum Height Calculator Helps You Find The Answer

Just relax and look how easy-to-use this maximum height calculator is:

Choose the velocity of the projectile. Let’s type 30 ft/s.

Enter the angle. Assume we’re kicking a ball at an angle of 70°.

Optionally, type the initial height. In our case, our starting position is the ground, so type in 0. Can the ball fly over a 13-ft fence?

Here it is – maximum height calculator displayed the answer! It’s 12.35 ft. So it will not fly over the mentioned barrier – throw it harder or increase the angle to reach your goal.

Just remember that we don’t take air resistance into account!

Objects Falling From Rest

Objects starting from rest have an initial velocity of zero, giving you your first kinematic quantity needed for problem solving. Beyond that, if you call the direction of initial motion positive, the object will have a positive acceleration and speed up as it falls.

An important first step in analyzing objects in free fall is deciding which direction along the y-axis you are going to call positive and which direction will therefore be negative. Although you can set your positive direction any way you want and get the correct answer, following the hints below can simplify your work to reach the correct answer consistently.

Identify the direction of the object’s initial motion and assign that as the positive direction. In the case of a dropped object, the positive y-direction will point toward the bottom of the paper.

With the axis identified you can now identify and write down your given kinematic information. Don’t forget that a dropped object has an initial velocity of zero.

v0=0

t ?

Answer: Vertical Problem: Declare down as the positive direction. This means that the acceleration, which is also down, is a positive quantity.

Read Also: Example Of Span Linear Algebra

Calculate The Time Of Flight:

The flight ends when the projectile object hits the surface of the earth. It happens when the vertical distance from the surface is 0. In the case where the initial height is 0, the formula can be written as: Vy * t g * t² / 2 = 0.

So, the equation used by maximum height calculator to find that the duration of flight is:

t = 2 * Vy / g =2 * V * sin / g.

However, if were projecting an object from some height, then the equation is not working so nicely as earlier. So, we get a quadratic equation to evaluate:

h + V_y * t g * t_2 / 2 = 0

After simplifying this equation, we get:

t = / g

Maximum Height Time Of Flight And Range Of A Projectile

The initial position of the projectile is at the origin O at t = 0. The projectile is launched with an initial velocity u at an angle , known as the angle of elevation, to the x-axis.

Its components in the x and y directions are:

ux = u cos

uy = u sin

Let ax and ay be the horizontal and vertical components, respectively, of the projectiles acceleration.

ax = 0

ay = g = 9.8 m s2

Horizontal Motion

vx = ux

x = uxt = u cos t

Vertical Motion

vy = uy gt = u sin gt

y = uyt ½gt2 = u sin t ½gt2

The horizontal motion is motion with constant velocity and the vertical motion is motion with constant acceleration.

What Is The Velocity Of The Ball Just Before Touching The Ground

And one important point, during the downward fall the magnitude of the velocity of the ball just before touching the ground would be the same as the magnitude of the velocity with which it was thrown upwards . We will prove it mathematically here:

Here while falling vertically downward, the ball falls the same height H

During this downward displacement, the initial velocity is equal to the final velocity of the upward movement i.e. v2. And we know that v2=0.

And during the downward movement, the final velocity is v3. This velocity is attained by the vertically falling ball just before touching the ground.

We have to find out the expression of this v3. Also, we have to find out the time taken for this downward fall.

v32 = v22 + 2 g H = 0 + 2 g = v12i.e. v3 = v1.

So we can say that during the downward fall the magnitude of the velocity of the ball just before touching the ground would be same as the magnitude of the velocity with which it was thrown upwards .