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?
Jennifer will compress the spring by a distance of 2.45 m.
To find out how far Jennifer will compress the spring, we can start by calculating the initial kinetic energy (KE) of Jennifer and the bobsled.
1. Calculate the initial kinetic energy (KE) using the equation: KE = 0.5 * mass * velocity^2
- The mass of Jennifer and the bobsled is 40.0 kg + 20.8 kg = 60.8 kg
- The velocity is given as 14.7 m/s
- KE = 0.5 * 60.8 kg * (14.7 m/s)^2
2. Calculate the gravitational potential energy (PE) when Jennifer and the bobsled are at the top of the ramp.
- The height (h) of the ramp can be calculated using the equation: h = length * sin(angle)
- The length is given as 46.0 m
- The angle (θ) is given as 21.0°
- h = 46.0 m * sin(21.0°)
- PE = mass * gravity * height
- Acceleration due to gravity (g) is approximately 9.8 m/s^2
3. Calculate the spring potential energy (PE) when the spring is at maximum compression.
- The spring potential energy (PE) is given by the equation: PE = 0.5 * k * x^2
- Where k is the spring constant, given as 1544.0 N/m
- And x is the compression distance we need to find.
To find the compression distance (x), we can equate the initial kinetic energy (KE) to the sum of the gravitational potential energy (PE) and the spring potential energy (PE). Therefore:
KE = PE + PE
0.5 * 60.8 kg * (14.7 m/s)^2 = mass * gravity * height + 0.5 * k * x^2
Simplifying and rearranging the equation, we can solve for x:
0.5 * 60.8 kg * (14.7 m/s)^2 - mass * gravity * height = 0.5 * k * x^2
Now, we can plug in the values and solve for x.
0.5 * 60.8 kg * (14.7 m/s)^2 - 60.8 kg * 9.8 m/s^2 * (46.0 m * sin(21.0°)) = 0.5 * 1544.0 N/m * x^2
Once you solve this equation, you will find the value of x, which represents how far Jennifer will compress the spring.
To find out how far Jennifer will compress the spring, we need to calculate the work done by Jennifer's kinetic energy as she slides down the ramp and into the spring. This work is equal to the potential energy stored in the compressed spring.
Let's break down the problem step by step:
1. Calculating the potential energy at the start of the ramp:
The potential energy at the start of the ramp is equal to the sum of Jennifer's gravitational potential energy and her initial kinetic energy.
Gravitational potential energy:
The height of the ramp is given by h = 46.0 m * sin(21.0°) since the ramp is sloped at an angle of 21.0°. Therefore, h = 15.769 m.
The gravitational potential energy is given by m * g * h, where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Plugging the values, potential energy at the start = 40.0 kg * 9.8 m/s^2 * 15.769 m = 6179.304 J.
2. Calculating the kinetic energy at the end of the ramp:
At the end of the ramp, Jennifer will have converted all her initial kinetic energy into potential energy stored in the compressed spring.
Since we're given Jennifer's maximum speed as 14.7 m/s, we can calculate her kinetic energy as (1/2) * m * v^2, where m is the mass and v is the velocity.
So, kinetic energy at the end = (1/2) * 40.0 kg * (14.7 m/s)^2 = 13716.6 J.
3. Calculating the compression distance using the potential energy:
The potential energy stored in a spring is given by (1/2) * k * x^2, where k is the spring constant and x is the compression distance.
Rearranging the formula, x^2 = (2 * potential energy) / k.
Plugging the values, x^2 = (2 * 13716.6 J) / 1544.0 N/m = 22.585 m^2.
Taking the square root of both sides, x = sqrt(22.585 m^2) = 4.749 m.
Therefore, Jennifer will compress the spring approximately 4.749 meters.