A new event has been proposed for the Winter Olympics. An athlete will sprint 150.0 m, starting from rest, then leap onto a 20.8 kg bobsled. The person and bobsled will then slide down a 46.0 m long ice-recovered ramp, sloped at á=21.0°, and into a spring with a carfully calibrated spring constant of 1544.0 N/m. The athlete who compresses the spring the farthest wins the gold medal. Jennifer, whose mass is 40.0 kg, has been training for this event. She can reach a maximum speed of 14.7 m/s in the 150.0 m dash. How far will Jennifer compress the spring?
Use a conservation of energy method.
Initial KE + gravitational PE loss = spring potential energy.
(1/2) (m + M) Vo^2 + (m + M) g H = (1/2) k X^2
Vo is the initial velocity after getting onto the bobsled. It is less than 14.7 m/s, because there will be some inelastic energy loss hopping on. You will have to use conservation of momentum to compute Vo.
m*(14.7) = Vo*(M+m)
H is the vertical distance her bobsled slides before hitting the spring, 46.0 m * sin 21.
m is her mass (40 kg), M is the bobsled mass (20.8 kg), and g is the acceleration of gravity.
Solve for X
To determine how far Jennifer will compress the spring, we need to calculate the work done on the spring. Work is equal to the potential energy stored in the spring, which is given by the equation W = (1/2) k x^2, where W is the work done, k is the spring constant, and x is the displacement or compression of the spring.
First, we need to determine the potential energy contribution from the sprinting part. This can be calculated using the kinetic energy formula: KE = (1/2) m v^2, where KE is the kinetic energy, m is the mass, and v is the velocity.
Given that Jennifer's mass is 40.0 kg and she reaches a maximum speed of 14.7 m/s, we can calculate her kinetic energy.
KE = (1/2) m v^2
KE = (1/2) (40.0 kg) (14.7 m/s)^2
KE = 1/2 * 40.0 * 215.49
KE ≈ 4309.8 J
Next, we need to determine the potential energy contribution from sliding down the ramp. The potential energy can be calculated using the formula: PE = m g h, where PE is the potential energy, m is the mass, g is the acceleration due to gravity, and h is the height.
Given that the ramp is sloped at an angle of 21.0° and the length of the ramp is 46.0 m, we can determine the height of the ramp using the trigonometric relation: h = L sin(θ), where L is the length and θ is the angle.
h = 46.0 m * sin(21.0°)
h ≈ 0.302 m
Now we can calculate the potential energy.
PE = m g h
PE = (40.0 kg) * (9.8 m/s^2) * (0.302 m)
PE ≈ 119.08 J
The total potential energy is the sum of the two contributions.
Total potential energy = KE + PE
Total potential energy = 4309.8 J + 119.08 J
Total potential energy ≈ 4428.88 J
Finally, we can determine the compression of the spring (x) using the equation: W = (1/2) k x^2.
4428.88 J = (1/2) (1544.0 N/m) x^2
Simplifying the equation:
8844.0 = 1544.0 x^2
Dividing both sides of the equation by 1544.0:
x^2 ≈ 5.72
Taking the square root of both sides of the equation:
x ≈ 2.39 meters
Therefore, Jennifer will compress the spring approximately 2.39 meters.