Math Problem Statement
A projectile is launched at an unknown angle from the top of an 85-m building. It lands a distance of 99 m from the base of the building 6.0 s after being thrown. What was its launch velocity?
Solution
To find the launch velocity of the projectile, we need to analyze the motion using two components: horizontal and vertical. Here's the breakdown of the problem:
1. Given Information:
- Height of the building (): 85 m
- Horizontal distance covered (): 99 m
- Time of flight (): 6.0 s
- Gravity ():
We need to find the initial velocity ().
2. Horizontal Motion:
The horizontal velocity () is constant throughout the motion, since there is no horizontal acceleration.
The formula for horizontal displacement is: Thus,
3. Vertical Motion:
The vertical motion is influenced by gravity. The formula for vertical displacement is: Where:
- is the vertical displacement, which is (since the projectile ends up 85 m below its initial position)
- is the initial vertical velocity (unknown)
Plugging in the values:
4. Total Launch Velocity:
The total launch velocity can be found using the Pythagorean theorem, since the initial velocity has both horizontal and vertical components:
Final Answer:
The launch velocity of the projectile is approximately 22.45 m/s.
Do you need more details on any step or have any questions?
Related Questions:
- What is the angle of launch given the initial velocity components?
- How would the launch velocity change if the time of flight was different?
- What would the horizontal velocity be if the projectile landed 120 m from the base?
- How long would the projectile be in the air if the height of the building was 100 m instead?
- How does the effect of air resistance alter the projectile's motion in real-life scenarios?
Tip:
Always break down projectile motion into horizontal and vertical components for simpler calculations!
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Math Problem Analysis
Mathematical Concepts
Projectile Motion
Kinematics
Trigonometry
Formulas
Horizontal displacement formula: d_x = v_{0x} * t
Vertical displacement formula: y = v_{0y} * t - (1/2) * g * t^2
Pythagorean theorem: v_0 = sqrt(v_{0x}^2 + v_{0y}^2)
Theorems
Pythagorean Theorem
Kinematic Equations
Suitable Grade Level
Grades 10-12
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