A little boy is standing at the edge of a cliff 1000 m high. He throws a ball straight downward at an initial speed of 20 m/s, and it falls straight down to the ground below. At a time of 6 seconds after it was thrown, how far above the ground is it? The acceleration due to gravity is 10 m/s2 .
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To determine the distance above the ground the ball is after 6 seconds, we can use the following kinematic equation to find the position:
s = ut + (1/2)at²,
where: - s is the distance traveled by the ball, - u is the initial velocity (20 m/s downward), - a is the acceleration due to gravity (10 m/s² downward), and - t is the time (6 seconds).
Since the ball is thrown downward with an initial speed, and gravity is also pulling it downward, both contributions to speed are in the same direction. Plugging in the values, the equation becomes:
s = (20 m/s)(6 s) + (1/2)(10 m/s²)(6 s)² s = 120 m + (1/2)(10)(36) s = 120 m + 5(36) s = 120 m + 180 m s = 300 m
The ball has traveled a distance of 300 meters downwards from the point it was thrown after 6 seconds.
Because the cliff is 1000 meters high, and the ball has moved 300 meters down, the distance above the ground (d) can be calculated using the initial height (H) of the cliff minus the distance (s) the ball has fallen:
d = H - s d = 1000 m - 300 m d = 700 m
After 6 seconds, the ball is 700 meters above the ground.
Extra: The kinematic equations are a set of four equations that describe the motion of objects under constant acceleration, with no other forces acting on the object except gravity. They are an essential part of Newtonian mechanics and are widely used in physics to predict the position, velocity, and acceleration of objects in motion.
The equation used in this scenario is particularly useful when we know the initial velocity and acceleration of an object, as well as the time it has been in motion. In this case, the motion was in one dimension (straight down) which simplifies the analysis significantly.
Remember that in kinematic equations, the direction of velocity and acceleration is critical. Since the ball was thrown downward, the initial velocity was taken as positive in the downward direction, and the acceleration due to gravity was also positive in the downward direction, which increases the overall speed of the ball as it continues to fall.
For someone studying physics or solving similar problems, it's crucial to keep in mind that choosing a consistent frame of reference and being careful with the signs of the values (positive or negative) depending on the direction is vital for obtaining the correct result.