An object is launched at a velocity of 20 m/s in a direction making an angle of 25° upward with the horizontal. a) What is the maximu

Question

An object is launched at a velocity of 20 m/s in a direction making an angle of 25° upward with the horizontal.
a) What is the maximum height reached by the object?
b) What is the total flight time (between launch and touching the ground) of the object?
c) What is the horizontal range (maximum x above ground) of the object?
d) What is the magnitude of the velocity of the object just before it hits the ground?

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Thu Nguyệt 4 weeks 2021-08-18T03:35:16+00:00 1 Answers 0 views 0

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    2021-08-18T03:36:21+00:00

    Answer:

    (a) max. height = 3.641 m

    (b) flight time = 1.723 s

    (c) horizontal range = 31.235 m

    (d) impact velocity = 20 m/s

    Above values have been given to third decimal.  Adjust significant figures to suit accuracy required.

    Explanation:

    This problem requires the use of kinematics equations

    v1^2-v0^2=2aS ………….(1)

    v1.t + at^2/2 = S …………(2)

    where

    v0=initial velocity

    v1=final velocity

    a=acceleration

    S=distance travelled

    SI units and degrees will be used throughout

    Let

    theta = angle of elevation = 25 degrees above horizontal

    v=initial velocity at 25 degrees elevation in m/s

    a = g = -9.81 = acceleration due to gravity (downwards)

    (a) Maximum height

    Consider vertical direction,

    v0 = v sin(theta) = 8.452 m/s

    To find maximum height, we find the distance travelled when vertical velocity = 0, i.e. v1=0,

    solve for S in equation (1)

    v1^2 – v0^2 = 2aS

    S = (v1^2-v0^2)/2g = (0-8.452^2)/(2*(-9.81)) = 3.641 m/s

    (b) total flight time

    We solve for the time t when the vertical height of the object is AGAIN = 0.

    Using equation (2) for vertical direction,

    v0*t + at^2/2 = S    substitute values

    8.452*t + (-9.81)t^2 = 3.641

    Solve for t in the above quadratic equation to get t=0, or t=1.723 s.

    So time for the flight = 1.723 s

    (c) Horiontal range

    We know the horizontal velocity is constant (neglect air resistance) at

    vh = v*cos(theta) = 25*cos(25) = 18.126 m/s

    Time of flight = 1.723 s

    Horizontal range = 18.126 m/s * 1.723 s = 31.235 m

    (d) Magnitude of object on hitting ground, Vfinal

    By symmetry of the trajectory, Vfinal = v = 20, or

    Vfinal = sqrt(v0^2+vh^2) = sqrt(8.452^2+18.126^2) = 20 m/s

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