A small steel ball bearing with a mass of 28.0 g is on a short compressed spring. When aimed vertically and suddenly released, the spring sends the bearing to a height of 1.25 m. Calculate the horizontal distance the ball would travel if the same spring were aimed 34.0 deg from the horizontal.

6.00

To calculate the horizontal distance the ball bearing would travel, we need to first find the initial velocity of the ball when it leaves the spring.

Using conservation of mechanical energy, we can equate the potential energy at the maximum height to the initial potential energy and kinetic energy when the ball is released.

The potential energy when the ball is at its maximum height is given by:
PE = mgh

Where:
m = mass of the steel ball bearing = 28.0 g = 0.028 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height = 1.25 m

PE = (0.028 kg)(9.8 m/s^2)(1.25 m)
PE = 0.343 J

Since the spring was compressed before releasing the ball, we know that some of the initial potential energy is stored as elastic potential energy.

The elastic potential energy (PE elastic) stored in a spring is given by:
PE elastic = (1/2)kx^2

Where:
k = spring constant
x = distance the spring is compressed

Since the spring constant (k) is not given, we cannot directly calculate the elastic potential energy. However, we can find the ratio of the potential energy (PE) to the elastic potential energy (PE elastic).

PE / PE elastic = 2

Substituting the known values:
0.343 J / PE elastic = 2

Solving for PE elastic:
PE elastic = 0.343 J / 2
PE elastic = 0.1715 J

Since the height at this angle (34.0 degrees from the horizontal) is different from the vertical release, the initial kinetic energy is also different.

The initial kinetic energy (KE) is given by:
KE = (1/2)mv^2

Where:
m = mass of the ball bearing = 0.028 kg
v = initial velocity

The total mechanical energy (ME) is defined by the sum of the potential energy (PE elastic) and the kinetic energy (KE).

ME = PE elastic + KE

Since mechanical energy is conserved, the total mechanical energy remains constant regardless of the angle at which the spring is released. Therefore, we can write:

ME = PE + KE

Substituting the known values:
0.1715 J + KE = 0.343 J

Solving for KE:
KE = 0.343 J - 0.1715 J
KE = 0.1715 J

Now, let's calculate the initial velocity (v) using the initial kinetic energy (KE):

0.1715 J = (1/2)(0.028 kg)v^2

Simplifying the equation:
0.343 J = (0.014 kg)v^2

Solving for v^2:
v^2 = 0.343 J / (0.014 kg)

v^2 = 24.5 m^2/s^2

v = √(24.5 m^2/s^2)
v ≈ 4.95 m/s

Now, to find the horizontal distance traveled, we can use the following kinematic equation for horizontal motion:

h = (v^2 / g) * sin(2θ)

Where:
h = horizontal distance
v = initial velocity = 4.95 m/s
g = acceleration due to gravity = 9.8 m/s^2
θ = angle = 34.0 degrees

Converting the angle to radians:
θ = 34.0 degrees * (π/180 radians/degree)
θ ≈ 0.593 radians

Substituting the known values and solving for h:
h = (4.95 m/s)^2 / (9.8 m/s^2) * sin(2 * 0.593 radians)
h ≈ 1.644 meters

Therefore, the horizontal distance the ball would travel would be approximately 1.644 meters.

To calculate the horizontal distance the ball would travel, we can use the conservation of mechanical energy. The initial energy stored in the compressed spring is converted into potential energy when the ball reaches its maximum height. This potential energy can then be used to calculate the maximum horizontal distance the ball will travel.

To begin, let's calculate the potential energy of the ball at its maximum height. The potential energy is given by the formula:

Potential energy (PE) = mass (m) * gravity (g) * height (h)

Given:
Mass (m) = 28.0 g = 0.028 kg (converted to kg)
Gravity (g) = 9.8 m/s^2
Height (h) = 1.25 m

PE = 0.028 kg * 9.8 m/s^2 * 1.25 m
= 0.3435 Joules (rounded to four decimal places)

Now, we can use the principle of conservation of mechanical energy to calculate the maximum horizontal distance (d) the ball will travel. The total mechanical energy of the system (spring and ball) is conserved, which consists of the potential energy at maximum height and the kinetic energy at launch.

Kinetic energy (KE) = Potential energy (PE)

The kinetic energy is given by the formula:

Kinetic energy (KE) = 1/2 * mass (m) * velocity^2

Since the ball is aimed at an angle, we need to consider the vertical and horizontal components of the velocity separately. The vertical velocity component is determined by the height reached, and the horizontal velocity component remains constant throughout the motion.

To find the vertical velocity (v_vertical), we can use the formula:

v_vertical = sqrt(2 * gravity * height)

v_vertical = sqrt(2 * 9.8 m/s^2 * 1.25 m)
= 6.279 m/s (rounded to three decimal places)

To find the horizontal velocity (v_horizontal), we need to use the angle provided (θ = 34.0 degrees). The horizontal component can be obtained using the formula:

v_horizontal = v_total * cos(θ)

v_horizontal = 6.279 m/s * cos(34.0 degrees)
= 5.20 m/s (rounded to two decimal places)

Now that we have the horizontal velocity, we can calculate the total flight time (t) of the ball using the formula:

t = 2 * h / v_vertical

t = 2 * 1.25 m / 6.279 m/s
= 0.399 seconds (rounded to three decimal places)

Finally, we can determine the maximum horizontal distance (d) traveled by the ball using the formula:

d = v_horizontal * t

d = 5.20 m/s * 0.399 s
= 2.074 meters (rounded to three decimal places)

Therefore, the ball would travel approximately 2.074 meters horizontally if the same spring were aimed at 34.0 degrees from the horizontal.