A rocket is launched with some initial velocity and travels with constant acceleration of 26.3m/s^2. When the rocket has reached a height of 170m above its launch position, its velocity is 197m/s. How long did it take the rocket to reach its height?

To solve for the time it took the rocket to reach a height of 170 meters, we can apply the kinematic equations of motion. The key equation that relates displacement (\( s \)), initial velocity (\( u \)), acceleration (\( a \)), and time (\( t \)) is:

\[
v = u + at
\]
\[
s = ut + \frac{1}{2} a t^2
\]

Where:
- \( v = 197 \, \text{m/s} \) is the final velocity,
- \( u \) is the initial velocity (which we do not know),
- \( a = 26.3 \, \text{m/s}^2 \) is the constant acceleration,
- \( s = 170 \, \text{m} \) is the displacement (height),
- \( t \) is the time (what we are solving for).

First, we can use the following kinematic equation that does not require time directly:

\[
v^2 = u^2 + 2as
\]

Solving for \( u \) (the initial velocity):

\[
u^2 = v^2 - 2as
\]

Substituting the given values:

\[
u^2 = (197)^2 - 2 \times 26.3 \times 170
\]

Let me calculate this.We found that \( u^2 = 29867 \), so the initial velocity \( u \) is:

\[
u = \sqrt{29867} \approx 172.88 \, \text{m/s}
\]

Now, using the first kinematic equation:

\[
v = u + at
\]

We can solve for \( t \):

\[
t = \frac{v - u}{a}
\]

Substituting the values:

\[
t = \frac{197 - 172.88}{26.3}
\]

I'll calculate the value of \( t \).The time it took the rocket to reach the height of 170 meters is approximately \( 0.92 \, \text{seconds} \).

Would you like more details or have any questions? Here are some related questions you might consider:

1. How would the time change if the acceleration were different?
2. What is the rocket’s velocity after 5 seconds of flight?
3. What would the displacement be after 3 seconds?
4. How long would it take to reach 500 meters height with the same conditions?
5. What is the significance of the initial velocity in determining the flight path?

**Tip:** Always check the units in kinematic problems to ensure consistency between velocity, acceleration, and time.

Learn how to calculate the time it takes for a rocket to reach a height of 170 meters when launched with constant acceleration of 26.3 m/s² and a final velocity of 197 m/s. This problem covers key kinematic equations and concepts, suitable for high school students in Grades 9-12.

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