A projectile is fired from the origin with an initial velocity V₁=100m/s at an angle θ₁=30⁰. Another is fired from a point on the x-axis at a distance x₀=60m from the origin with initial velocity V₂=80m/s at an angle of θ₂. It is desired to have the two projectiles collide at some point P(x,y).
A. )What is θ₂?
B.) At what time and point in space do they collide?
C.) Find velocity components of each just before impact.
t=3
The projectile fired from the origin impacts 883.67m downrange after reaching a height of 127.55m.
The second projectile will collide with the first projectile before it reaches its maximum height if the second projectile's launch angle was selected to result in equal maximum heights.
Without another boundary condition, there are a multitude of launch angles for the second projectile that will result in an intercept of the first.
Might there be another piece of information that would open the solution window.
For instance:
Assume that the second projectile is launched such that it has the same maximum altitude as the first.
We make use of the two equations for height and distance of projectiles:
h = Vo^2(sin(a)^2)/2g and
d = Vo^2(sin(2b))/g
where
h = the maximum height reached – meters
d = the horizontal distance traveled - m
Vo = the initial launch velocity – meters/sec.
a and b = the angle of the velocity vectors to the horizontal
g = the acceleration due to gravity - 9.8m/sec.^2
Given:
Projectile 1………Projectile 2
Vo = 100m/s…...……80m/s
a = 30 deg
b = ……………..….....TBD
g = 9.8m/sec.^2……9.8
100^2(sin30)^2/2(9.8 = 80^2(siny)^2/2(9.8 or y = 38.682 deg.
d1 = 100^2(sin60)/9.8 = 883.67m
d2 = 80^2(sin77.364)/9.8 = 637.24m
Time of intercept
100(cos30)t = 60 + 80(cos938.682)t or t = 2.48 sec.
Point of intercept
x = 100(.866)2.48 = 214.76m
y = 50(2.48) – 4.9(2.48)^2 = 93.86m.
I’ll let you figure out the velocity info.
Note that if the maximum height of the second projectile was allowed to be any quantity, several answers are possible.
To solve this problem, we need to break it down into smaller steps. Let's go through each question one by one:
A. What is θ₂?
To find θ₂, we can use the information provided and apply some trigonometry. Since the projectile is fired from a point on the x-axis, we know that its initial vertical velocity component, V₂y, is 0. The initial horizontal velocity component, V₂x, can be determined by using the given initial velocity (V₂ = 80 m/s) and the angle θ₂. From there, we can use trigonometric functions to find θ₂.
B. At what time and point in space do they collide?
To determine when and where the two projectiles collide, we need to analyze their motion independently and find the time it takes for each projectile to reach point P(x,y). We'll start by finding the time it takes for each projectile to reach point P.
For the first projectile (fired with V₁ = 100 m/s and θ₁ = 30⁰), we can split the initial velocity into horizontal and vertical components (V₁x and V₁y). Then, we can calculate the time it takes for the vertical position (y-coordinate) of this projectile to reach zero (since both projectiles must collide at y = 0).
For the second projectile (fired with V₂ = 80 m/s and θ₂), we'll follow a similar process by splitting its initial velocity into horizontal and vertical components (V₂x and V₂y). We want to find the time it takes for this projectile to travel the horizontal distance x₀ (60 m) plus some additional distance x to reach point P.
To find the time of collision, we equate the times for both projectiles to reach point P. Solving the resulting equation will give us the time at which they collide. Once we have the time, we can substitute it back into either projectile's equations to find the x and y coordinates of the collision point.
C. Find velocity components of each just before impact.
To find the velocity components of each projectile just before impact, we can use their equations of motion and substitute the time of impact into these equations to find their velocities at that specific time.
By following these steps, we can find the answers to all three questions and determine the values of θ₂, the collision time and point, as well as the velocity components just before impact for both projectiles.