You and a friend stand on a snow-covered roof. You both throw snowballs from an elevation of 18 m with the same initial speed of 11 m/s, but in different directions. You through your snowball downward, at 40° below the horizontal; your friend throws her snowball upward, at 40° above the horizontal. What is the speed of each ball when it is 5.0 m above the ground? (Neglect air resistance.)
To find the speed of each ball when it is 5.0 m above the ground, we can use the principles of projectile motion. The motion of the objects can be divided into horizontal and vertical components.
First, let's analyze the motion of the snowball thrown downwards by you. We can call this object Snowball A. The initial speed of Snowball A is 11 m/s, and it is thrown at an angle of 40° below the horizontal.
To find the vertical component of its velocity, we can use the formula:
Vy = V * sin(θ)
Where Vy is the vertical component of velocity, V is the initial speed (11 m/s), and θ is the angle below the horizontal (40°). Plugging in the values, we get:
Vy_A = 11 m/s * sin(40°)
Next, we can use the formula to find the time taken for Snowball A to reach the height of 5.0 m:
Δy = Vyt + (1/2) * g * t^2
Where Δy is the change in vertical position (5.0 m), Vy is the initial vertical velocity (Vy_A), g is the acceleration due to gravity (9.8 m/s^2), and t is the time taken. Rearranging the equation, we have:
5.0 m = Vy_A * t + (1/2) * g * t^2
Solving this equation will give us the time taken by Snowball A to reach 5.0 m above the ground.
Now let's analyze the motion of the snowball thrown upwards by your friend. We can call this object Snowball B. The initial speed of Snowball B is also 11 m/s, but it is thrown at an angle of 40° above the horizontal.
Using the same method as above, we can find the vertical component of velocity for Snowball B:
Vy_B = 11 m/s * sin(40°)
Next, we can set up the equation to find the time taken by Snowball B to reach a height of 5.0 m:
5.0 m = Vy_B * t - (1/2) * g * t^2
Finally, we can plug in the values for Vy_A, Vy_B, and solve the respective equations for t_A and t_B (the times taken by Snowball A and Snowball B to reach 5.0 m), using the quadratic formula.
Once we have the times, we can determine the speeds of the snowballs at that height by multiplying the times by the horizontal velocities. Since the initial horizontal velocity is the same for both snowballs (11 m/s), the final speed will be the same for both snowballs.
So, the speed of each ball when it is 5.0 m above the ground will be 11 m/s.