A new event has been proposed for the Winter Olympics. As seen in the gure, an a

Question

A new event has been proposed for the Winter Olympics. As seen in the gure, an athlete will sprint
100 m, starting from rest, then leap onto a 20 kg bobsled. The person and bobsled will then slide
down a 50-m-long ice-covered ramp, sloped at 20, and into a spring with a carefully calibrated spring
constant of 2000 N/m. The athlete who compresses the spring the farthest wins the gold medal. Lisa,
whose mass is 40 kg, has been training for this event. She can reach a maximum speed of 12 m/s in
the 100 m dash.

(a) How far will Lisa compress the spring?
(b) The Olympic committee has very exact speci cations about the shape and angle of the ramp. Is
this necessary? What factors about the ramp are important?

show steps please~

Explanation / Answer

a)

First we need to find the speed of Lisa and the bobsled as soon as she jumps on. This can be considered a completely inelastic collision, since she stays on the bobsled. Use the conservation of momentum equation:
m1v1i + m2v2i = (m1 + m2)vf
(40 kg)(12 m/s) + (20 kg)(0 m/s) = (40 + 20 kg)vf
vf = 8.0 m/s

Now we can find how much Lisa and the bobsled will compress the spring. This problem is easiest to solve using conservation of energy with springs:
(KE + PEg + PEs)i = (KE + PEg + PEs)f
We can cross out PEsi, KEf, and PEgf because those are zero.
KEi + PEgi = PEsf
1/2 mv^2 + mgh = 1/2 kx^2
sin(20°) = opp/hyp = h/(50 m)
h = (50 m)sin(20°) = 17 m
1/2 (60 kg)(8.0 m/s)^2 + (60 kg)(9.8 m/s^2)(17 m) = 1/2 (2000 N/m)x^2
x = 3.5 m

b)

The reason that i find the ACTUAL height is becasue the steeper or the higher than angle of the ramp, the more energy lisa has to compress the spring. The length of the ramp, and the angle matters, because the bigger and longer the ramp, the more energy you have to push the spring.

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