Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [projectile-motion]

This tag is for questions regarding to "projectile motion", the motion of an object thrown or projected into the air, subject to only the acceleration of gravity.

When a particle is thrown obliquely near the earth’s surface, it moves along a curved path under constant acceleration that is directed towards the center of the earth (we assume that the particle remains close to the surface of the earth).The object is called a projectile, its path is called its trajectory and the motion is called projectile motion.

Example: The motion of falling objects, as covered in Problem-Solving Basics for One-Dimensional Kinematics, is a simple one-dimensional type of projectile motion in which there is no horizontal movement.

  • In a projectile motion, the only acceleration acting is in the vertical direction which is acceleration due to gravity $(g)$.
  • Air resistance to the motion of the body is to be assumed absent in projectile motion.
  • Projectile motion only occurs when there is one force applied at the beginning on the trajectory, after which the only interference is from gravity.
  • Equations of motion can be applied separately in $X$-axis and $Y$-axis to find the unknown parameters.

For more details follow the references.

References:

https://en.wikipedia.org/wiki/Projectile_motion

https://byjus.com/physics/projectile-motion/

https://courses.lumenlearning.com/physics/chapter/3-4-projectile-motion/

220 questions
18
votes
6 answers

What is minimum speed needed to jump over sphere object that has radius R and at distance d?

(I am not expert in English. I will write as well as I can.) To understand this question easier, lets see this picture. From this picture, what is minimum initial speed that this grasshopper need to jump over this log? The grasshopper movement path…
Ro Theory
  • 695
  • 1
  • 6
  • 18
16
votes
7 answers

Envelope of Projectile Trajectories

For a given launch velocity $v$ and launch angle $\theta$, the trajectory of a projectile may be described by the standard formula $$y=x\tan\theta-\frac {gx^2}{2v^2}\sec^2\theta$$ For different values of $\theta$ what is the envelope of the…
Hypergeometricx
  • 22,229
  • 2
  • 30
  • 83
7
votes
2 answers

Ideal angle to launch from a swing to maximize distance

When I was little (and even now if I can get the chance) I liked to play on swings, and my favorite method of dismounting was to let go mid-swing and fly thru the air. That got me wondering what the best point is to let go of a swing to maximize the…
WB-man
  • 2,428
  • 12
  • 22
5
votes
2 answers

What is the optimal angle to kick a ball in rugby game?

What is the best angle to kick a ball toward the other team such that when two teams run at each other the teams will meet (when the kicking team tackles the team with the ball) at the distance furthest back from the ball kicker as possible? The way…
Tom
  • 157
  • 9
5
votes
1 answer

Do these balls collide?

Assume that two balls $B_1,B_2$ of radius $r$ continuously move around inside of a square of size $d$. They bounce off the walls, i.e. the $x$-component of the velocity is multiplied with $-1$ when they hit the left/right wall, and similarly for the…
Martin Brandenburg
  • 154,019
  • 16
  • 267
  • 487
5
votes
3 answers

Are Position and Velocity (or Velocity and Acceleration) Vectors Always Parallel?

While reading Chapter 1 of an astrodynamics textbook, I came across the statement: $$\mathbf{v}\cdot \mathbf{{\dot{v}}}=v{\dot{v}}$$ In other words, the dot product of velocity and the time-rate-of-change of velocity is simply equal to the product…
Bryson
  • 53
  • 6
4
votes
4 answers

Projectile: $v^*w^*=gk$ for minimum launch velocity

A projectile launched from $O(0,0)$ at velocity $v$ and launch angle $\theta$, passes through $P(k,h)$. The velocity of the projectile at $P$ is $w$. The slope of $OP$ is $\alpha$, i.e. $\tan\alpha=\frac hk$, and the length of $OP$ is $R$.…
Hypergeometricx
  • 22,229
  • 2
  • 30
  • 83
4
votes
2 answers

Projectile Motion with Differential Equations

I've been trying to write code that will calculate the required intercept angles of a projectile launch to hit a moving target, a person running around a game world. I have no issues programming things, but I need a bit of help with the…
John Adams
  • 43
  • 4
4
votes
2 answers

High School Projectile Motion and Quadratics

High school students are learning about the basics of solving quadratics and trigonometric ratios, including trigonometric inverses. The eventual goal of their project is to be able to show a reasonable firing solution, given in initial angle…
Zediiiii
  • 167
  • 6
4
votes
2 answers

Do springs and projectiles under gravity exist in the same family?

Background A 1D vertical spring subject to gravity satisfies Hook's Law: $m x''(t) = -kx(t) + g$ where $m$ is the mass at the end of the spring, $x(t)$ is the position of the mass at time $t$, $x''(t)$ is its acceleration, $k$ is the stiffness…
Alec Jacobson
  • 484
  • 2
  • 13
3
votes
1 answer

Finding Projectile Angle With Different Elevation When Velocity And Range Are Known

I'm trying to derive a formula to find the angle $θ$ required to hit a target that may be higher or lower than the initial launch position. My known variables are $g, v_0, y_0, y_f, x_0, x_f$ where: $θ$ is the initial launch angle and the variable…
Utaru
  • 33
  • 3
2
votes
1 answer

Ball bouncing off a function

Say we have a ball that is dropped from a point $(a,h) $ and falls under acceleration $g$ onto a function $f(x)$, where it hits the function at the point $(a,f(a))$, $h>f(a)$. It will then follow a parabolic path that is defined by the gradient at…
Sick Nutmeg
  • 481
  • 9
2
votes
1 answer

Inquiry on Orbital Mechanics

I'm an undergraduate (currently studying precalculus) with an interest in physics. Specifically, I'm interested in the relationship between physics and math which seems to occur often. For example, it is my understanding that Newton conceived of…
muldrowhanif
  • 21
  • 1
2
votes
1 answer

Formulae for projectile motion with resistance proportional to velocity

What are the formulae for resisted projectile motion in which the resistance is proportional to the velocity? I have a problem where my answers don't match up with the textbook answers and I need to verify with formulae. I need the velocities and…
Waleed Dahshan
  • 524
  • 2
  • 13
2
votes
2 answers

maximum range of projectile launched from elevation

I am trying to find the maximum range of projectile from an elevation. I found the answer in this question, but I have two questions: Why does $y$ need to be $0?$ Why do they differentiate with respect to $\theta,$ and what is the meaning of this?
Ahmed mk
  • 41
  • 5
1
2 3
…
14 15