Question

A ball was tossed from a height of 1.20 m, initial velocity of 13.0 m/s at an angle 26.0° above the horizontal. The ball hits the ground at a horizontal distance of 20.0 m from the launch point. Use 10.0 N/kg for g.

a) Find in m/s the magnitude of the velocity of the ball when it is at the top most position of its trajectory

b) Find in m/s² the magnitude of the acceleration of the ball when it is at the top most position of its trajectory.

c) How long in seconds is the ball in flight?

d) What is in m/s the magnitude of the velocity with which the ball hits the ground?

e) What is in m the highest point relative to ground reached by the ball?

Answer 1

Y_o = initial position = 1.20 m

V_o = initial velocity = 13 m/s

V_ox = initial velocity along X-direction = V_o Cos = 13 Cos26 = 11.7 m/s

V_oy = initial velocity along Y-direction = V_o Sin = 13 Sin26 = 5.7 m/s

X = horizontal distance travelled = 20 m

t = time of travel

a)

At the topmost position, V_fy = y-component of velocity = 0 (since the ball comes to a momentary rest)

At the topmost position, V_fx = X-component of velocity = V_ox = 11.7 m/s      (since velocity along X-direction remains constant)

So net velocity at the topmost point = V_t = sqrt (V_fy² + V_fx²) = sqrt (11.7² + 0²) = 11.7 m/s

b)

at topmost position , the acceleration is due to gravity in downward direction, so a_y = 10 m/s²

c)

t = time of travel

Consider the motion along X-direction

X = V_ox t

t = X / V_ox = 20 / 11.7 = 1.71 sec

d)

Consider the motion along Y-direction

V_fy = V_oy + at

V_fy = 5.7 + (-10) (1.71)

V_fy = - 11.4 m/s

Consider the motion along X-direction

V_fx = V_ox = 11.7 m/s

net velocity as the ball hits the ground

V = sqrt (V_fx² + V_fy²) = sqrt (11.7² + (-11.4)²)

V = 16.34 m/s

e)

at highest point , vertical velocity becomes 0 , so V_fy =0

consider the motion along Y-direction

V_oy = 5.7 m/s

a = - 10 m/s²

Y_o = initial position of launch point = 1.20 m

Y = height gained above the ground

Using the equation

V_fy² = V_ox² + 2 a (Y-Y_o )

0² = 5.7² + 2 (-10) (Y-Y_o )

(Y-Y_o ) = 2.825

Y = 2.825 + 1.20

Y = 4.025 m