Math Problem Statement
An artillery shell is fired with an initial velocity of 300 m/s at 54.0° above the horizontal. To clear an avalanche, it explodes on a mountainside 40.5 s after firing. What are the x- and y-coordinates of the shell where it explodes, relative to its firing point?
Solution
To solve for the - and -coordinates of the shell at the point of explosion, we need to break down the motion into its horizontal () and vertical () components using the initial velocity and the time of flight.
Given:
- Initial velocity () = 300 m/s
- Launch angle () = 54.0°
- Time of flight () = 40.5 s
- Acceleration due to gravity () = 9.8 m/s²
Step 1: Resolve initial velocity into horizontal and vertical components.
The initial velocity can be split into two components:
- Horizontal component:
- Vertical component:
Step 2: Calculate the horizontal displacement (-coordinate).
The horizontal displacement is given by:
Since there is no acceleration in the horizontal direction (ignoring air resistance), the horizontal velocity remains constant.
Step 3: Calculate the vertical displacement (-coordinate).
The vertical displacement is affected by gravity, and the equation for vertical displacement is:
Now, let's compute these components step by step. I'll calculate the values for you.The coordinates of the shell at the point where it explodes are:
- -coordinate: 7141.6 meters
- -coordinate: 1792.3 meters
These are the horizontal and vertical distances from the firing point at the moment of the explosion.
Would you like further details or have any questions?
Here are 5 related questions:
- How would air resistance affect the projectile's trajectory?
- How would a different launch angle impact the range and height?
- What would be the maximum height the shell reaches during its flight?
- How would the time of flight change if the launch velocity were different?
- How do we calculate the impact velocity of the shell?
Tip: When solving projectile motion problems, it's helpful to break down the motion into horizontal and vertical components, treating each independently.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Projectile Motion
Kinematics
Trigonometry
Formulas
v_{0x} = v_0 \cos(\theta)
v_{0y} = v_0 \sin(\theta)
x = v_{0x} \cdot t
y = v_{0y} \cdot t - \frac{1}{2} g \cdot t^2
Theorems
Equations of Motion
Suitable Grade Level
Grades 10-12
Related Recommendation
Projectile Motion: Missile Fired at 300 m/s at 30°
Calculate Horizontal and Vertical Velocities for a Cannonball Launched at 70 Degrees
Projectile Motion: Calculate Velocities, Time, Height, and Range
Projectile Motion of a Cannonball: Maximum Height and Range
Projectile Motion: Find the Range of a Projectile Shot from a Height