A boy 14.0m above the ground in a tree throws a ball for his dog, who is standing right below the tree and starts running the instant the ball is thrown. If the boy throws the ball upward at 50.0∘ above the horizontal, at 8.00m/s .
To analyze this scenario, we need to break it down into a few steps:
Step 1: Determine the initial vertical velocity of the ball.
Step 2: Calculate the time it takes for the ball to reach the ground.
Step 3: Calculate the horizontal distance the dog covers in that time.
Step 1: Determine the initial vertical velocity of the ball.
Since the boy throws the ball upward, the initial vertical velocity will be in the positive direction. The vertical component of the velocity is given by:
v_y = v * sin(theta)
where v is the initial velocity of the ball (given as 8.00 m/s) and theta is the angle above the horizontal (given as 50.0°). Let's calculate v_y:
v_y = 8.00 m/s * sin(50.0°)
Step 2: Calculate the time it takes for the ball to reach the ground.
To calculate the time it takes for the ball to reach the ground, we can use the vertical motion equation:
y = y_0 + v_0 * t + (1/2) * a * t^2
where y is the vertical position, y_0 is the initial vertical position (14.0 m above the ground), v_0 is the initial vertical velocity (v_y), a is the acceleration due to gravity (-9.8 m/s^2), and t is the time. We want to find the time it takes for the ball to reach the ground, so when the ball reaches the ground, y will be zero. We can rearrange the equation to solve for t:
0 = 14.0 m + (8.00 m/s * sin(50.0°)) * t + (1/2) * (-9.8 m/s^2) * t^2
Simplifying this equation will give us a quadratic equation in terms of t. We can use the quadratic formula to solve for t.
Step 3: Calculate the horizontal distance the dog covers in that time.
Once we have the time it takes for the ball to reach the ground, we can calculate the horizontal distance the dog covers. The horizontal component of the velocity remains constant throughout the motion. We can use the equation:
x = x_0 + v_0_x * t
where x is the horizontal distance covered, x_0 is the initial horizontal position (which is zero since the dog starts right below the tree), v_0_x is the initial horizontal velocity, and t is the time. Since the ball is thrown at an angle above the horizontal, we can calculate the initial horizontal velocity using:
v_0_x = v * cos(theta)
where v is the initial velocity (given as 8.00 m/s) and theta is the angle above the horizontal (given as 50.0°). We can then plug in the values in the equation to calculate the horizontal distance.