A 500 g model rocket is on a cart that is rolling to the right at a speed of. The rocket engine, when it is fired, exerts an 8.0 N vertical thrust on the rocket. Your goal is to have the rocket pass through a small horizontal hoop that is 20 m above the ground. At what horizontal distance left of the hoop should you launch?

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Convert the mass of the rocket from grams to kilograms: \( m = 500 \ \text{g} = 0.5 \ \text{kg} \). This is necessary because the SI unit of mass is kilograms.

Determine the vertical acceleration of the rocket due to the thrust force. Use Newton's second law: \( F = ma \), where \( F \) is the thrust force (8.0 N) and \( m \) is the mass of the rocket. Solve for \( a \): \( a = \frac{F}{m} \).

Calculate the time it takes for the rocket to reach the height of 20 m. Use the kinematic equation for vertical motion: \( y = \frac{1}{2} a t^2 \), where \( y \) is the vertical displacement (20 m), \( a \) is the vertical acceleration from step 2, and \( t \) is the time. Solve for \( t \): \( t = \sqrt{\frac{2y}{a}} \).

Determine the horizontal velocity of the rocket. Since the cart is rolling to the right, the rocket's horizontal velocity is the same as the cart's initial velocity. Let this velocity be \( v_x \).

Calculate the horizontal distance the rocket travels during the time \( t \). Use the equation \( x = v_x t \), where \( x \) is the horizontal distance, \( v_x \) is the horizontal velocity, and \( t \) is the time calculated in step 3. Subtract this distance from the position of the hoop to find the horizontal distance left of the hoop where the rocket should be launched.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and is subject to gravitational forces. It can be analyzed in two dimensions: horizontal and vertical. The horizontal motion is uniform, while the vertical motion is influenced by gravity, leading to a parabolic trajectory. Understanding this concept is crucial for determining the launch angle and distance needed for the rocket to reach the hoop.

Thrust and Forces

Thrust is the force exerted by the rocket engine to propel the rocket upward. In this scenario, the thrust of 8.0 N acts vertically against the force of gravity, which affects the rocket's vertical ascent. Analyzing the net forces acting on the rocket helps in calculating its vertical acceleration and the time it takes to reach the desired height of 20 m, which is essential for timing the launch correctly.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. In this problem, these equations will be used to calculate the time it takes for the rocket to ascend to the hoop's height and the horizontal distance it travels during that time, allowing for the determination of the optimal launch position.

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