A ball is thrown horizontally from a 90 m cliff and strikes the ground 70 m from the base, what is the initial velocity?
1 answer:
Answer:
The initial velocity is 16 m/s
Explanation:
The position of the ball at a time t is given by the following vector:
r = (x0 + v0x · t, y0 + v0y · t + 1/2 · g · t²)
Where:
r = position vector at time t
x0 = initial horizontal position
v0x = initial horizontal velocity
t = time
y0 = initial vertical position
v0y = initial vertical velocity
g = acceleration due to gravity (-9.8 m/s² considering the upward direction as positive)
When the ball reaches the ground its position vector will be "r final" (see attached figure). The x-component of this vector is 70 m and the y-component is -90 m. Notice that the origin of the frame of reference is located at the throwing point (y0 and x0 = 0). Then, we can use the equations for the x and y-component of "r" to obtain the time of flight and the initial velocity:
Using the equation for the y-component of the vector position:
y = y0 + v0y · t + 1/2 · g · t² (y0 and v0y = 0)
-90 m = -1/2 · 9.8 m/s² · t²
-2 · -90 m/ 9.8 m/s² = t²
t = 4.3 s
Using the equation for the x-component of the vector position:
x = x0 + v0x · t
70 m = v0x · 4.3 s
v0x = 16 m/s
The initial velocity is 16 m/s
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Answer:
5.24 m/s
Explanation:
So for the rod to be able to rise upward to the straight up position, the kinetic energy caused by linear speed v0 must be just enough to convert into the potential energy.
Since the rod is uniform in mass, we can treat the body as 1 point, at its center of mass, or geometric center, aka 0.35 / 2 = 0.175 m from the pivot.
For the rod to swing from bottom to top, the center must have moved a distance of h = 0.175 * 2 = 0.35 m, vertically speaking.
Since we neglect friction and air resistance, according to the law of energy conservation then:
Where v is the speed at the center of mass, g = 9.81 m/s2 is the gravitational acceleration, and m is the mass. We can divide both sides by m
As this is only the speed at the center of mass, the speed at the bottom end would be different, to calculate this, we need to find the common angular speed:
Where r is the rotation radius, or the distance from pivot point to the center of mass
Where R is the distance from the pivot to the bottom end of the rod
- Mass=m=1kg
- Height=h=2cm=0.02m
- Acceleration due to gravity=g=10m/s^2