To determine the point at which the two balls will collide, we need to find the time it takes for each ball to reach that point. Let's analyze the motion of each ball separately and find the time of collision.
For the ball that is dropped vertically from a height of 10m: We can use the equation of motion for free fall: h = (1/2)gt^2, where h is the height, g is the acceleration due to gravity, and t is the time.
Since the ball is dropped, its initial velocity (u) is 0, and the initial position (h) is 10m. Plugging in these values into the equation, we have: 10 = (1/2)(9.8)t^2. Simplifying the equation: t^2 = 2(10/9.8), t^2 = 20/9.8, t^2 ≈ 2.04, t ≈ √2.04 ≈ 1.43s.
Therefore, it will take approximately 1.43 seconds for the dropped ball to hit the ground.
For the ball thrown vertically upward with an initial velocity of 10m/s: To find the height at which the two balls will collide, we need to determine the time it takes for the ball to reach its highest point. At the highest point, the ball's vertical velocity will become 0, and it will start falling back down.
We can use the equation of motion for vertical motion: v = u + gt, where v is the final velocity, u is the initial velocity, g is the acceleration due to gravity, and t is the time.
At the highest point, the final velocity (v) is 0, the initial velocity (u) is 10m/s, and the acceleration due to gravity (g) is -9.8m/s^2 (negative because it acts in the opposite direction to the initial velocity). Plugging in these values into the equation, we have: 0 = 10 - 9.8t. Simplifying the equation: 9.8t = 10, t ≈ 10/9.8 ≈ 1.02s.
Therefore, it will take approximately 1.02 seconds for the thrown ball to reach its highest point.
Since the thrown ball takes less time to reach its highest point than the dropped ball takes to hit the ground, the collision will occur during the upward motion of the thrown ball. To find the height at which the two balls collide, we can use the equation of motion for the thrown ball:
h = ut + (1/2)gt^2,
where h is the height, u is the initial velocity, g is the acceleration due to gravity, and t is the time. We need to find the height at time t = 1.02s, when the thrown ball is at its highest point.
Plugging in the values: h = (10)(1.02) - (1/2)(9.8)(1.02)^2, h ≈ 10.2 - 5.02, h ≈ 5.18m.
Therefore, the two balls will collide at a height of approximately 5.18 meters above the ground.