Assume that an intercontinental ballistic missile goes from rest to a suborbital speed of 6.50 km/s in 60.0 s (the actual speed and time are classified). What is its average acceleration in m/s² and in multiples of g (9.80 m/s²)?
The Correct Answer and Explanation is :
Solution:
To calculate the average acceleration, we will use the formula:
[
a_{\text{avg}} = \frac{\Delta v}{\Delta t}
]
Where:
- ( a_{\text{avg}} ) is the average acceleration,
- ( \Delta v ) is the change in velocity,
- ( \Delta t ) is the change in time.
Given values:
- Initial velocity ( v_0 = 0 \, \text{m/s} ) (since the missile starts from rest),
- Final velocity ( v_f = 6.50 \, \text{km/s} = 6500 \, \text{m/s} ),
- Time ( t = 60.0 \, \text{s} ).
Step 1: Calculate the change in velocity
The change in velocity ( \Delta v ) is simply the final velocity minus the initial velocity:
[
\Delta v = v_f – v_0 = 6500 \, \text{m/s} – 0 \, \text{m/s} = 6500 \, \text{m/s}
]
Step 2: Calculate the average acceleration
Now we substitute the values into the formula for average acceleration:
[
a_{\text{avg}} = \frac{\Delta v}{\Delta t} = \frac{6500 \, \text{m/s}}{60.0 \, \text{s}} = 108.33 \, \text{m/s}^2
]
Step 3: Express the average acceleration in terms of multiples of ( g )
To convert this to multiples of ( g ) (where ( g = 9.80 \, \text{m/s}^2 )):
[
a_{\text{avg}} \, (\text{in multiples of} \, g) = \frac{108.33 \, \text{m/s}^2}{9.80 \, \text{m/s}^2} \approx 11.05 \, g
]
Final Answer:
- The average acceleration is ( 108.33 \, \text{m/s}^2 ).
- The average acceleration is approximately ( 11.05 \, g ).
Explanation:
The missile accelerates from rest to a speed of 6.50 km/s in 60.0 seconds. By dividing the total change in velocity by the time it takes to achieve this speed, we obtain the average acceleration, which represents the rate at which the missile’s speed is increasing. Expressing this value in terms of ( g ) gives a sense of how intense the acceleration is compared to the force of gravity on Earth. A value of 11.05 ( g ) is significant because it indicates that the missile’s acceleration is more than 11 times the force of gravity acting on an object on Earth’s surface.
Now, I’ll generate a visual representation of the calculation.
Here is an illustration of the missile’s acceleration, showing the change in speed and the comparison with Earth’s gravity. This visual represents the missile’s average acceleration and its comparison to ( 11.05g ). Let me know if you’d like further clarification or additional details!
