You have a 2.4 m high platform that is level on top. A spring launcher is located on the top so that it launches objects horizontally. You want to launch a 0.70 kg ball so that it hits at target on the ground 6.0 m away from the base. If the spring constant is 500 N/m, what distance must you compress the spring?
To find the distance you need to compress the spring, you can use the principles of energy conservation. The initial potential energy stored in the compressed spring will be transferred to the ball as kinetic energy when it is launched. This kinetic energy will then convert into both horizontal and vertical components.
First, let's calculate the gravitational potential energy of the ball when it is at the platform.
Gravitational potential energy (PE) = mass (m) x acceleration due to gravity (g) x height (h)
PE = 0.70 kg x 9.8 m/s^2 x 2.4 m
PE = 16.296 J
Next, let's calculate the horizontal velocity (V_h) of the ball when it hits the target. Assuming there is no air resistance, the horizontal component of velocity remains constant throughout the flight.
The horizontal distance (d) traveled by the ball is given as 6.0 m. The time taken (t) to travel this distance can be found using the equation:
d = V_h x t
Therefore,
t = d / V_h
Now, let's calculate the final vertical position (h_f) of the ball when it hits the target. Using the equation of motion, specifically the one for vertical displacement:
h_f = h_i + V_iy * t + 0.5 * a_y * t^2
Since the vertical velocity (V_iy) at the start is 0 and the acceleration due to gravity (a_y) is -9.8 m/s² (downward direction), the equation simplifies to:
h_f = h_i - 4.9 * t^2
Plugging in the known values, we get:
0 = 2.4 - 4.9 * (d / V_h)^2
Now, let's consider the kinetic energy when the ball is launched. The spring potential energy (PE_spring) transferred to the ball is given by:
PE_spring = 0.5 * k * x^2
Where k is the spring constant, and x is the compression distance of the spring.
Therefore, the initial kinetic energy (KE_initial) of the ball is:
KE_initial = PE_spring
KE_initial = 0.5 * k * x^2
The final kinetic energy (KE_final) of the ball when it hits the target can be calculated as:
KE_final = 0.5 * m * V_h^2
By applying the principle of conservation of energy, we can equate these two expressions:
0.5 * k * x^2 = 0.5 * m * V_h^2
Now, we can substitute the value of V_h using the previously calculated time (t) and horizontal distance (d):
0.5 * k * x^2 = 0.5 * m * (d / t)^2
Finally, let's solve for x:
x^2 = (m * d^2) / (k * t^2)
x = √((m * d^2) / (k * t^2))
Substituting the given values into the equation will give you the distance you need to compress the spring.