A small steel ball bearing with a mass of 10 g is on a short compressed spring. When aimed vertically and suddenly released, the spring sends the bearing to a height of 1.19 m.
Calculate the speed at which the ball leaves the spring.
Calculate the horizontal distance the ball will travel if the same spring is aimed 37° from the horizontal.
To calculate the speed at which the ball bearing leaves the spring, we can use the principle of conservation of mechanical energy. The total mechanical energy of the system remains constant, neglecting any losses due to friction or air resistance.
Step 1: Calculate the potential energy at the topmost point of the motion.
At the highest point, the potential energy is solely due to the vertical displacement of the ball bearing. We can determine the potential energy using the equation:
Potential energy (PE) = mass (m) * acceleration due to gravity (g) * height (h)
Here, the mass of the ball bearing is 10 g, which is equivalent to 0.01 kg. The acceleration due to gravity is approximately 9.8 m/s^2, and the height reached is 1.19 m.
Plugging in these values, we get:
PE = 0.01 kg * 9.8 m/s^2 * 1.19 m = 0.11542 J
Step 2: Calculate the initial kinetic energy at the moment of release.
At the moment of release, the potential energy is fully converted into kinetic energy. Therefore, we can equate the potential energy to the initial kinetic energy:
Kinetic energy (KE) = Potential energy (PE)
So,
KE = 0.11542 J
Step 3: Calculate the speed of the ball bearing.
The kinetic energy is given by the equation:
KE = 0.5 * mass (m) * velocity^2
Rearranging the formula to solve for velocity (v), we have:
v = √(2 * KE / m)
Substituting the values, we obtain:
v = √(2 * 0.11542 J / 0.01 kg) ≈ 1.36 m/s
Hence, the speed at which the ball bearing leaves the spring is approximately 1.36 m/s.
To calculate the horizontal distance the ball will travel when aimed at 37° from the horizontal, we need to analyze the projectile motion of the ball bearing.
Step 1: Determine the initial vertical velocity.
We can calculate the initial vertical velocity (v_y) using the equation:
v_y = v * sin(θ)
Here, v represents the speed of the ball bearing (1.36 m/s), and θ is the launch angle (37°).
Plugging in these values, we can find:
v_y = 1.36 m/s * sin(37°) ≈ 0.82 m/s
Step 2: Determine the initial horizontal velocity.
The initial horizontal velocity (v_x) remains constant throughout the motion and can be calculated using the equation:
v_x = v * cos(θ)
Substituting the given values:
v_x = 1.36 m/s * cos(37°) ≈ 1.09 m/s
Step 3: Calculate the time of flight.
The total time of flight (t) can be determined from the equation:
t = 2 * (v_y) / g
Substituting the values:
t = 2 * 0.82 m/s / 9.8 m/s^2 ≈ 0.167 s
Step 4: Calculate the horizontal distance traveled.
The horizontal distance (d) can be calculated using the equation:
d = v_x * t
Substituting the values:
d = 1.09 m/s * 0.167 s ≈ 0.182 m
Hence, the ball will travel approximately 0.182 m horizontally when aimed 37° from the horizontal.