A small steel ball bearing with a mass of 22 g is on a short compressed spring. When aimed vertically and suddenly released, the spring sends the bearing to a height of 1.21 m.
Calculate the horizontal distance the ball will travel if the same spring is aimed 32degrees from the horizontal.
To calculate the horizontal distance the ball will travel, we need to consider the projectile motion of the ball. Projectile motion consists of two independent motions: horizontal motion and vertical motion. We can find the horizontal distance by calculating the horizontal component of the ball's velocity and the time it takes for the ball to reach its maximum height.
Let's break down the problem step by step:
Step 1: Calculate the initial vertical velocity (Vy0):
The ball is launched vertically, so there is no horizontal motion initially. We can use the equation of motion for vertical displacement to find the initial vertical velocity:
Vf^2 = Vi^2 + 2 * g * Δy,
where Vf is the final velocity, Vi is the initial velocity, g is the acceleration due to gravity (approximately 9.8 m/s^2), and Δy is the vertical displacement (1.21 m).
At the highest point, the final velocity is zero (Vf = 0) since the ball momentarily stops before falling down. Thus, the equation becomes:
0 = Vi^2 + 2 * 9.8 * 1.21.
Simplifying and solving for Vi, we have:
Vi = sqrt(2 * 9.8 * 1.21).
Step 2: Calculate the horizontal initial velocity (Vx0):
The horizontal component of the ball's velocity remains constant throughout the motion. We can find the initial horizontal velocity using the angle at which the ball is launched (32 degrees) and the magnitude of the initial velocity (Vi):
Vx0 = Vi * cos(θ),
where θ is the launch angle (32 degrees).
Step 3: Calculate the time of flight (t):
To calculate the horizontal distance, we need to know the time it takes for the ball to reach its maximum height and then fall back to the ground. The time of flight can be found using the equation for vertical motion:
Vy = Vy0 + g * t,
where Vy is the vertical velocity at any given time t. At the highest point, the vertical velocity is zero (Vy = 0). We can rearrange the equation to solve for the time of flight:
0 = Vi * sin(θ) - g * t.
Simplifying and solving for t, we have:
t = Vi * sin(θ) / g.
Step 4: Calculate the horizontal distance (D):
Finally, we can calculate the horizontal distance traveled by the ball. Since the horizontal velocity (Vx0) is constant and the time of flight (t) represents the total time of the projectile motion, we can use the formula:
D = Vx0 * t.
Substituting the values we calculated earlier:
D = (Vi * cos(θ)) * (Vi * sin(θ) / g).
Now you can substitute the values of Vi, θ, and g into the equation to find the solution.