3.A rocket launched accelerates at 3.5m/s^2 in 5.90 secs and2.98m/s^2 in the next 5.98 secs and then
experiences a free fall. What time will the rocket be in air?
Assume that the rocket is launched from the ground.
To find the total time the rocket will be in the air, we need to calculate the time it takes for the rocket to reach its maximum acceleration and then add the time it takes for the rocket to experience free fall.
First, let's find the time it takes for the rocket to reach its maximum acceleration.
Given:
Initial acceleration (a1) = 3.5 m/s^2
Time taken to reach maximum acceleration (t1) = 5.90 s
We can use the formula:
Final velocity (v) = Initial velocity (u) + (acceleration x time)
As the rocket starts from rest, the initial velocity is 0 m/s. Hence, we can rewrite the formula as:
v1 = u + (a1 x t1)
v1 = 0 + (3.5 x 5.90)
v1 = 20.65 m/s
Now, let's find the time it takes for the rocket to experience free fall.
Given:
Acceleration due to gravity (a2) = 9.8 m/s^2
Time taken for free fall (t2) = ?
We can use the formula:
Final velocity (v) = Initial velocity (u) + (acceleration x time)
As the rocket is experiencing free fall, the initial velocity is v1 (the final velocity before free fall). Hence, we can rewrite the formula as:
0 = v1 + (a2 x t2)
t2 = -v1 / a2
t2 = -20.65 / 9.8
t2 ≈ -2.11 s
Since time cannot be negative in this context, we can ignore the negative sign.
Now, let's calculate the total time the rocket will be in the air:
Total time = t1 + t2
Total time = 5.90 + 2.11
Total time ≈ 7.01 s
Therefore, the rocket will be in the air for approximately 7.01 seconds.
To find the total time the rocket will be in the air, we need to calculate the time it takes for the rocket to accelerate and the time it takes for the rocket to free fall.
First, let's calculate the time it takes for the rocket to accelerate. The initial acceleration is 3.5 m/s^2. We will use the equation of motion:
Final velocity = Initial velocity + (Acceleration × Time)
In this case, the initial velocity is 0 m/s (since the rocket starts from rest), and the final velocity is unknown. We can rearrange the equation to solve for time:
Time = (Final velocity - Initial velocity) / Acceleration
Using the given acceleration of 3.5 m/s^2, we can calculate the time for the first phase:
Time1 = (Final velocity - 0) / 3.5 m/s^2
Time1 = Final velocity / 3.5 m/s^2
Next, let's calculate the time it takes for the rocket to accelerate during the second phase. The initial acceleration for this phase is 2.98 m/s^2, and the final velocity is still unknown. Following the same steps as before:
Time2 = Final velocity / 2.98 m/s^2
Now we need to find the total time of acceleration by adding Time1 and Time2:
Total acceleration time = Time1 + Time2
After the acceleration phase, the rocket experiences free fall. In free fall, the rocket will fall with an acceleration of approximately 9.8 m/s^2. To find the time during free fall, we can use the equation:
Time3 = sqrt(2 × Distance / Acceleration)
Since the distance is not given in the problem, we can assume that the rocket will reach a maximum height and fall back to the ground. The distance in this case will be twice the maximum height the rocket reaches during the acceleration phase.
Total free fall time = Time3
Finally, we can calculate the total time the rocket will be in the air by adding the total acceleration time and the total free fall time:
Total time in air = Total acceleration time + Total free fall time
Plug in the values, solve the equations, and you'll find the time the rocket will be in the air.