a rocket is launched at an angle of 53 degrees above the horizontal with an initial speed of 75m/s. It moves for 25 seconds along its line of motion with an overall acceleration of 25 m/s^2. What is its maximum height?
First, let us find the vertical and horizontal components of the initial velocity.
The vertical component of the initial velocity (Vv0) can be calculated as follows:
Vv0 = Vi * sin(θ)
where:
Vi = initial speed = 75 m/s
θ = angle of projection = 53°
Vv0 = 75 * sin(53°)
Vv0 ≈ 60 m/s
Next, let's find the acceleration. The given acceleration is along the line of motion at the 53-degree angle, so we have to find the vertical component of the acceleration.
The vertical component of the acceleration (Av) can be calculated as follows:
Av = Atotal * sin(θ)
where:
Atotal = overall acceleration = 25 m/s²
θ = angle of projection = 53°
Av = 25 * sin(53°)
Av ≈ 20 m/s²
Now we have the initial vertical velocity Vv0 and vertical acceleration Av. We can find the time taken to reach the maximum height.
At the maximum height, the vertical velocity will be 0. We can use the following formula to find the time taken:
Vvf = Vv0 + a * t
where:
Vvf = final vertical velocity = 0
Vv0 = initial vertical velocity = 60 m/s
a = acceleration = -20 m/s² (negative value because it opposes the motion)
t = time taken to reach maximum height
Solving for t:
0 = 60 - 20 * t
t = 60 / 20
t = 3 seconds
Now that we have the time taken to reach the maximum height, we can find the maximum height using the following formula:
h = Vv0 * t + 0.5 * a * t²
where:
Vv0 = initial vertical velocity = 60 m/s
t = time taken to reach maximum height = 3 seconds
a = acceleration = -20 m/s²
h = maximum height
h = 60 * 3 + 0.5 * (-20) * 3²
h = 180 - 0.5 * 20 * 9
h = 180 - 90
h = 90 meters
The rocket's maximum height is 90 meters.
To find the rocket's maximum height, we need to break down the motion into horizontal and vertical components.
First, let's find the time it takes for the rocket to reach its maximum height. Since the rocket moves for 25 seconds along its line of motion and the overall acceleration affects only the horizontal component, the time it takes for the rocket to reach its maximum height will be the same as the total time elapsed, which is 25 seconds.
Next, let's find the vertical velocity component at the maximum height. We can use the equation of motion:
v = u + at
where:
v = final velocity (vertical component)
u = initial vertical velocity
a = acceleration (in the vertical direction)
t = time
We know that the initial vertical velocity is zero (as it starts from the ground), the acceleration due to gravity is -9.8 m/s^2 (as gravity acts in the downward direction), and the time is 25 seconds.
Using these values, we can calculate the final vertical velocity:
v = 0 + (-9.8 m/s^2) * 25 s
v = -245 m/s
Note: The negative sign indicates that the velocity is directed in the opposite direction to the vertical motion (downward).
Now, let's find the maximum height using the equation of motion:
h = u*t + (1/2)*a*t^2
where:
h = maximum height
u = initial vertical velocity
t = time
a = acceleration (in the vertical direction)
We know that the initial vertical velocity is zero (as it starts from the ground), the acceleration in the vertical direction is -9.8 m/s^2 (as gravity acts in the downward direction), and the time is 25 seconds.
Using these values, we can calculate the maximum height:
h = 0 + (1/2)*(-9.8 m/s^2)*(25 s)^2
h = (1/2)*(-9.8 m/s^2)*(625 s^2)
h = (-4.9 m/s^2)*(625 s^2)
h = -3062.5 m^2/s^2
The negative sign indicates that the height is in the opposite direction to the vertical motion (below the starting point).
Therefore, the maximum height of the rocket is approximately 3062.5 meters below the starting point.
To find the maximum height reached by the rocket, we can use the equations of motion.
Step 1: Find the vertical component of the initial velocity.
The initial velocity can be broken down into horizontal and vertical components using basic trigonometry. The vertical component can be found by multiplying the initial speed by the sine of the launch angle.
Vertical component of initial velocity = initial speed * sin(launch angle)
Vertical component of initial velocity = 75 m/s * sin(53°)
Step 2: Find the time taken to reach the maximum height.
The time taken to reach the maximum height can be found using the formula:
time = (final velocity - initial velocity) / acceleration
Since the rocket is moving upwards against the force of gravity, the final velocity at the maximum height is 0 m/s. Therefore, the time taken to reach the maximum height is:
time = (0 - vertical component of initial velocity) / acceleration
time = (- vertical component of initial velocity) / acceleration
time = (-75 m/s * sin(53°)) / (-9.8 m/s²)
Step 3: Calculate the maximum height.
The maximum height can be calculated using the formula:
maximum height = initial vertical velocity * time + (1/2) * acceleration * time²
maximum height = (vertical component of initial velocity) * time + (1/2) * acceleration * time²
maximum height = (75 m/s * sin(53°)) * [(-75 m/s * sin(53°)) / (-9.8 m/s²)] + (1/2) * (-9.8 m/s²) * [(-75 m/s * sin(53°)) / (-9.8 m/s²)]²
By substituting the values and simplifying, you can find the maximum height reached by the rocket.