Projectile motion but with resistive force

[tubworld]

I have this question:

A 10kg projectile is launched with an initial speed of 100m/s at an elevation of 35 degrees. The resistive force is R = -bv , where b =10kg/s.

Determine the horizontal and vertical component coordinates of the projectile as functions of time.

How do I do this?? I urgently need the complete solution to this! Thanx!

 

[LeonhardEuler]

Basically you're looking for a differential equation for the x-component of the velocity. This is not difficult to find because you can work with this component without thinking about the y-component. You know that $\vec{R}=-b\vec{v}$, so then $R_x=-bv_x$. You also know that $\vec{F}=\vec{R}=m\vec{a}=m\vec{\dot{v}}$, so $R_x=m\dot{v_x}$. Substitute for R. Do you recognize the solution of the differential equation you get?

 

[quicknote]

I am happy to help you with this question. Projectile motion with resistive force can be a bit more complex, but it can be solved using the equations of motion and some basic principles of physics.

First, let's define the variables in the problem:
- m = 10kg (mass of the projectile)
- v0 = 100m/s (initial velocity)
- θ = 35° (launch angle)
- b = 10kg/s (resistive force coefficient)

To determine the horizontal and vertical components of the projectile's motion as functions of time, we can use the following equations:

Horizontal displacement (x):
x = v0*cos(θ)*t

Vertical displacement (y):
y = v0*sin(θ)*t - (1/2)*g*t^2

Where g is the acceleration due to gravity, which is approximately 9.8m/s^2.

Now, let's consider the resistive force. We know that the resistive force is equal to R = -bv, where v is the velocity of the projectile. This means that as the projectile moves, the resistive force will act in the opposite direction of its motion, slowing it down.

To incorporate the resistive force into our equations, we need to modify the acceleration term. Since acceleration is the rate of change of velocity, we can write it as:

a = dv/dt

Substituting the resistive force equation, we get:

a = (-b/m)*v

This is known as the differential equation for resistive force. To solve it, we need to use calculus. However, since you need an urgent solution, I will provide you with the final equations for the horizontal and vertical components of the projectile's motion as functions of time:

Horizontal displacement (x):
x = (v0*cos(θ)/b)*(1 - e^(-b*t/m))

Vertical displacement (y):
y = (v0*sin(θ)/b)*(1 - e^(-b*t/m)) - (m*g/b)*t

Note that these equations assume that the projectile is launched at t=0. Also, e is the mathematical constant e = 2.71828... (known as Euler's number).

I hope this helps you understand how to solve this problem. Keep in mind that these equations are simplified and do not take into account other factors such as air resistance, which can also affect the motion of