A 360 g model rocket is on a cart that is rolling to the right at a speed of 2.5 m/s. The rocket engine, when it is fired, exerts an 8.5 N thrust on the rocket. Your goal is to have the rocket pass through a small horizontal hoop that is 20 m above the launch point. At what horizontal distance left of the loop should you launch?
ayy lmao
To determine the horizontal distance left of the loop at which you should launch the rocket, we need to consider the forces acting on the rocket and calculate the trajectory of the rocket's flight.
Here's how we can approach this problem step by step:
Step 1: Analyze the forces acting on the rocket before and after the engine is fired.
Before the engine is fired:
- The rocket is initially at rest in the cart, so it experiences only the force of gravity acting vertically downward.
After the engine is fired:
- The rocket experiences the force of gravity acting downward.
- The rocket engine exerts a thrust force horizontally to the right.
Step 2: Determine the net force on the rocket.
To find the net force, we need to calculate the vertical and horizontal components of the forces.
Vertical forces:
- The force of gravity acting downward (weight) can be calculated using the formula: weight = mass x gravitational acceleration.
Given: mass = 0.360 kg (360 g), gravitational acceleration = 9.8 m/s².
Weight = 0.360 kg x 9.8 m/s² = 3.528 N
Horizontal forces:
- Since the only force acting horizontally is the rocket engine's thrust, we can directly use the given value of 8.5 N.
Net Force:
- The net force acting on the rocket is the vector sum of the vertical and horizontal forces.
- Since the forces act perpendicular to each other, we can use Pythagoras' theorem to find the net force magnitude: net force = √(vertical force² + horizontal force²).
Net force = √(3.528 N² + 8.5 N²) = √(12.384 N²) = 3.519 N
Step 3: Calculate the acceleration of the rocket.
- The acceleration of the rocket can be determined using Newton's second law: force = mass x acceleration.
- Since the net force acting on the rocket is the only horizontal force, we can calculate acceleration using the horizontal force value.
Acceleration = Net force / mass = 3.519 N / 0.360 kg = 9.774 m/s²
Step 4: Determine the time taken for the rocket to reach the hoop.
To calculate the time, we'll use the vertical motion of the rocket since we're interested in the height of the hoop.
- The vertical distance the rocket needs to travel is 20 m.
- The initial vertical velocity is zero (since the rocket starts from rest).
- The acceleration due to gravity is 9.8 m/s² (acting downward).
- We can use the second equation of motion: s = ut + (1/2)at², where s is the vertical distance, u is the initial velocity, a is the acceleration, and t is the time.
20 m = (0)t + (1/2)(-9.8 m/s²)t²
10t² = 20 m
t² = 2 s²
t = √2 s ≈ 1.41 s
Step 5: Calculate the horizontal distance the rocket travels during that time.
To determine the horizontal distance the rocket covers during this time, we'll use kinematic equations since the initial velocity in the horizontal direction is given.
- The horizontal velocity of the rocket is 2.5 m/s (since the cart is rolling).
- The time taken for the rocket to reach the hoop is approximately 1.41 seconds.
- We can use the equation: distance = velocity x time.
Distance = 2.5 m/s x 1.41 s ≈ 3.525 m
Therefore, to pass through the small horizontal hoop 20 m above the launch point, you should launch the rocket approximately 3.525 meters left of the hoop.