For a science fair competition, a group of High School students built a kicker-machine that can launch a golf ball from the origin with a velocity of 12.9 m/s and initial angle of 31.7° with respect to the horizontal.
a) Where will the golf ball fall back to the ground?
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b) How high will it be at the highest point of its trajectory?
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c) What is the ball's velocity vector (in Cartesian components) at the highest point of its trajectory? First enter x, then y.
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d) What is the ball's acceleration vector (in Cartesian components) at the highest point of its trajectory? First enter x, then y.
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a) To determine where the golf ball will fall back to the ground, we can use the kinematic equations of motion. The horizontal and vertical motions of the ball can be treated independently.
The horizontal motion remains at a constant velocity throughout, so the time it takes for the ball to fall back to the ground will be the same as the time it takes for the ball to reach its maximum height. This can be found using the equation:
time = (2 * vertical component of initial velocity) / (acceleration due to gravity)
Substituting the given values:
time = (2 * 12.9 m/s * sin(31.7°)) / 9.8 m/s^2
Now, we can use the time to find the horizontal distance traveled by the ball using the equation:
horizontal distance = horizontal component of initial velocity * time
Substituting the given values:
horizontal distance = 12.9 m/s * cos(31.7°) * time
This will give us the horizontal distance from the origin where the ball will fall back to the ground.
b) To find the maximum height reached by the golf ball, we can use the equation for vertical projectile motion:
maximum height = (vertical component of initial velocity)^2 / (2 * acceleration due to gravity)
Substituting the given values:
maximum height = (12.9 m/s * sin(31.7°))^2 / (2 * 9.8 m/s^2)
This will give us the maximum height of the ball at the highest point of its trajectory.
c) At the highest point of its trajectory, the vertical component of the ball's velocity will be 0, while the horizontal component will remain constant. Therefore, the velocity vector at the highest point will have its x-component equal to the initial horizontal velocity and its y-component equal to 0.
d) At the highest point of its trajectory, the ball's acceleration vector will have its x-component equal to 0, since there is no force acting horizontally. The y-component of the acceleration will be equal to the acceleration due to gravity, directed downward.