Math Problem Statement
A projectile is fired with an initial speed of 38.0 m/sm/s at an angle of 42.7 ∘∘ above the horizontal on a long flat firing range. Determine the speed of the projectile 1.50 ss after firing.
Solution
To find the speed of the projectile 1.50 seconds after firing, we need to break down the initial velocity into horizontal and vertical components, then determine the velocity at that time using the kinematic equations.
Step 1: Break the initial velocity into components
Given:
- Initial speed
- Angle
- Time
- Acceleration due to gravity
We need to resolve the initial velocity into horizontal and vertical components:
Step 2: Calculate the horizontal and vertical components
Using trigonometric values:
Now, calculate the components:
Step 3: Determine the velocity components after 1.50 seconds
Horizontal component ():
Since there is no horizontal acceleration (ignoring air resistance), the horizontal velocity remains constant:
Vertical component ():
The vertical velocity changes due to gravity. The formula to calculate the vertical velocity after time is:
Substitute the values:
Step 4: Calculate the magnitude of the velocity (speed)
Now that we have both the horizontal and vertical components of the velocity at , we can calculate the resultant speed using the Pythagorean theorem:
Final Answer:
The speed of the projectile 1.50 seconds after firing is approximately 30.1 m/s (rounded to three significant figures).
Would you like more details or have any questions? Here are five related questions for further practice:
- What is the horizontal displacement of the projectile after 1.50 seconds?
- What is the maximum height reached by the projectile?
- How long does the projectile stay in the air before hitting the ground?
- What is the total horizontal distance traveled by the projectile (its range)?
- How fast is the projectile moving at its maximum height?
Tip: When calculating projectile motion, always separate the motion into horizontal and vertical components and treat them 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(θ)
v_{0y} = v_0 * sin(θ)
v_y = v_{0y} - g * t
v = sqrt(v_x^2 + v_y^2)
Theorems
Pythagorean Theorem
Kinematic Equations
Suitable Grade Level
Grades 9-12
Related Recommendation
Calculate Horizontal Component of Initial Velocity for Projectile Launched at 32 m/s
Projectile Motion: Speed of a Fired Projectile with Initial Height and Angle
Find Launch Components of a Projectile at 30 Degrees
Calculate Maximum Height of a Projectile Fired at 37.9 m/s at 43.4°
Projectile Motion with Initial Speed of 80 m/s at 35 Degrees