What is the maths behind projectiles (including air resistance)?

In summary: We have Newton's second law:\mathbf{F} = m\frac{d^2\mathbf{r}}{dt^2}Where r = xi + yj is position, where x=x(t) and y=y(t). Splitting into vertical and horizontal components:-mg -\kappa y^\prime = my^{\prime\prime}-\kappa x^\prime = mx^{\prime\prime}Where \kappa = 1/15. In canonical form:y^{\prime\prime} + \frac{\kappa}{m} y^\prime = gx^{\prime\prime
  • #1
Georgepowell
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E.g. A football of mass 1kg is thrown with a speed of 10m/s at an angle of 30 degrees from a horizontal plane. Take air resistance (in Newtons) as 1/15 th of the speed of the ball in the exact opposite direction that the ball is traveling. How long is the ball in the air? and how far (horizontally) has it traveled at the point of landing?

I understand how to figure it out if I can ignore air resistance, but how would I answer this question?

I am not so much interested in the answer, but more how to figure out the answer, I have just given you that example so it gives you something to answer if you need an example.
 
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  • #3
That tells me how to figure out the air resistance, and not how to predict its movement after I know it.

Anybody else?
 
  • #4
Georgepowell said:
That tells me how to figure out the air resistance, and not how to predict its movement after I know it.

Anybody else?
One would need to formulate a system of differential equations using Newton's second law. The exercise is fairly straight forward for the case of linear drag (as you have described in your opening post).
 
  • #5
Ok, so from what I already know I can write the instantaneous acceleration in terms of the velocity at that time, but I can't seem to figure out how to have acceleration/velocity as a function of time.

Any help on this?
 
  • #6
Georgepowell said:
Ok, so from what I already know I can write the instantaneous acceleration in terms of the velocity at that time, but I can't seem to figure out how to have acceleration/velocity as a function of time.

Any help on this?
We have Newton's second law:

[tex]\mathbf{F} = m\frac{d^2\mathbf{r}}{dt^2}[/tex]

Where r = xi + yj is position, where x=x(t) and y=y(t). Splitting into vertical and horizontal components:

[tex]-mg -\kappa y^\prime = my^{\prime\prime}[/tex]

[tex]-\kappa x^\prime = mx^{\prime\prime}[/tex]

Where [itex]\kappa = 1/15[/itex]. In canonical form:

[tex]y^{\prime\prime} + \frac{\kappa}{m} y^\prime = g[/tex]

[tex]x^{\prime\prime} + \frac{\kappa}{m} x^\prime = 0[/tex]

Both of the above are second order differential equations with constant coefficients. Both are separable and straightforward to solve.
 
  • #7

1. What is the equation for the trajectory of a projectile?

The equation for the trajectory of a projectile is y = y0 + xtanθ - (gx2)/(2v02cos2θ), where y0 is the initial height, x is the horizontal distance, θ is the launch angle, g is the acceleration due to gravity, and v0 is the initial velocity.

2. How does air resistance affect the path of a projectile?

Air resistance, also known as drag, is a force that acts in the opposite direction of the projectile's motion. It increases as the projectile's speed increases and can significantly alter the trajectory of the projectile. As a result, the equation for the trajectory of a projectile is adjusted to include the drag force, making the path more parabolic than the usual idealized parabola.

3. What is the role of air density in projectile motion?

Air density plays a crucial role in determining the amount of air resistance acting on a projectile. The higher the air density, the greater the drag force, and the more significant the effect on the projectile's trajectory. Air density is affected by factors such as altitude, temperature, and humidity.

4. How is the launch angle of a projectile chosen to maximize distance?

The launch angle that maximizes the distance of a projectile depends on the specific conditions, such as the initial velocity and air resistance. In general, the optimal launch angle is around 45 degrees, as it combines the effects of horizontal and vertical velocities to achieve the maximum range. However, a detailed analysis of the specific variables is necessary to determine the exact launch angle for maximum distance.

5. Can air resistance ever be neglected in projectile motion calculations?

In most real-life scenarios, air resistance cannot be neglected in projectile motion calculations as it significantly affects the trajectory of the projectile. However, in certain situations, such as when the projectile's speed is very low or when the distance traveled is short, the effects of air resistance may be negligible and can be ignored in calculations.

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