To determine the instantaneous acceleration of the rocket when it is fired, we need to consider both the upward acceleration provided by the rocket and the downward acceleration due to gravity. The rocket is released at an angle of 20° to the vertical, which means we will need to break down the forces acting on it into their vertical and horizontal components.
Understanding the Forces at Play
The rocket experiences two main accelerations:
- The upward acceleration of the rocket, which is 8 m/s².
- The downward gravitational acceleration, which is 9.1 m/s².
Breaking Down the Accelerations
Since the rocket is launched at an angle, we can analyze the vertical component of its acceleration. The angle given is 20° to the vertical, which means the angle to the horizontal is 70° (90° - 20°). However, for our calculations, we will focus on the vertical components.
Calculating the Vertical Component of the Rocket's Acceleration
The vertical component of the rocket's acceleration can be calculated using trigonometric functions. Since the upward acceleration is directed at an angle, we can find the vertical component using the cosine function:
Vertical Component of Rocket's Acceleration:
A_vertical = A_rocket * cos(θ)
A_vertical = 8 m/s² * cos(20°)
Using a calculator, we find that:
A_vertical ≈ 8 m/s² * 0.9397 ≈ 7.52 m/s²
Net Acceleration Calculation
Now, we need to consider the net acceleration acting on the rocket. Since the gravitational acceleration acts downward, we subtract it from the upward component of the rocket's acceleration:
Net Acceleration:
A_net = A_vertical - g
A_net = 7.52 m/s² - 9.1 m/s²
A_net ≈ -1.58 m/s²
The negative sign indicates that the net acceleration is directed downward, meaning that the force of gravity is greater than the upward force provided by the rocket's acceleration at that moment.
Final Thoughts
In summary, the instantaneous acceleration of the rocket when it was fired is approximately -1.58 m/s². This negative value signifies that, despite the rocket's initial upward thrust, gravity is pulling it down more strongly than the rocket can accelerate upward at that angle. Understanding these forces is crucial in analyzing the motion of objects in flight.
