A rocket is launched at an angle of 59.0° above the horizontal with an initial speed of 103 m/s. The rocket moves for 3.00 s along its initial line of motion with an acceleration of 31.0 m/s2. At this time, its engines fail and the rocket proceeds to move as a projectile.
(a) Find the maximum altitude reached by the rocket.
m
(b) Find its total time of flight.
s
(c) Find its horizontal range.
m
To find the maximum altitude reached by the rocket, we need to determine the vertical component of its initial velocity and the time it takes for the rocket to reach its maximum height.
(a) Finding the vertical component of the initial velocity:
The vertical component of the initial velocity can be found using the equation:
Viy = Vi * sin(θ)
where,
Viy is the vertical component of the initial velocity,
Vi is the initial speed of the rocket, and
θ is the launch angle.
Plugging in the given values:
Vi = 103 m/s
θ = 59.0°
Viy = 103 m/s * sin(59.0°) = 88.67 m/s
(b) Finding the time it takes to reach maximum altitude:
The time it takes for the rocket to reach its maximum height can be determined using the equation:
Vfy = Viy + a * t
where,
Vfy is the final y-component of velocity (0 m/s at maximum height),
Viy is the initial y-component of velocity (88.67 m/s),
a is the acceleration in the y-direction (-9.8 m/s^2, considering the gravitational acceleration),
and t is the time taken to reach the maximum height.
Rearranging the equation:
0 m/s = 88.67 m/s + (-9.8 m/s^2) * t
-88.67 m/s = -9.8 m/s^2 * t
t = -88.67 m/s / -9.8 m/s^2 = 9.05 s
(Note: We consider -9.8 m/s^2 because the acceleration due to gravity is directed downwards.)
The time it takes to reach the maximum height is 9.05 s.
To find the maximum altitude, we can use the kinematic equation:
Δy = Viy * t + (1/2) * a * t^2
where,
Δy is the maximum altitude (unknown),
Viy is the initial y-component of velocity (88.67 m/s),
a is the acceleration in the y-direction (-9.8 m/s^2), and
t is the time taken to reach the maximum height (9.05 s).
Plugging in the values:
Δy = 88.67 m/s * 9.05 s + (1/2) * (-9.8 m/s^2) * (9.05 s)^2
Δy = 400.06 m
Therefore, the maximum altitude reached by the rocket is 400.06 m.
(c) Finding the horizontal range:
The horizontal range can be determined using the equation:
R = Vix * t
where,
R is the horizontal range (unknown),
Vix is the initial x-component of velocity,
and t is the total time of flight.
To find Vix, we use the equation:
Vix = Vi * cos(θ)
where,
Vi is the initial speed of the rocket,
and θ is the launch angle.
Plugging in the given values:
Vi = 103 m/s
θ = 59.0°
Vix = 103 m/s * cos(59.0°) = 52.42 m/s
To find the total time of flight, we can use the equation:
t = 2 * t1
where,
t is the total time of flight,
and t1 is the time taken to reach maximum height (9.05 s).
Plugging in the values:
t = 2 * 9.05 s
t = 18.1 s
Finally, we can find the horizontal range:
R = Vix * t
R = 52.42 m/s * 18.1 s
R = 948.46 m
Therefore, the horizontal range of the rocket is 948.46 m.