Anew event has been proposed for the winter olympics. 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 degrees , 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. how far x will lisa compress the spring?

3.5 m

Explanation:

Given

Mass of the bob sled , m_b = 20 kg

Mass of the girl, m = 40 kg

Speed of the girl, v = 12 m/s

Spring constant , k = 2000 N/m

Length of the ramp, L = 50 m

Angle of incline , θ = 20°

When the girl leaps on the sled , we use conservation of momentum principle

to find the speed of sled and the girl

m v + m_b (0) = (m + m_b) v_1

40 kg ×12 m/s = (20 kg + 40 kg ) v_1

v_1= 8 m/s

The height of the incline is ,

h = L sin 20 = 50 m  sin 20° = 17.1 m

By law of conservation of energy

(m + m_b) g h + 1/2 (m + m_b) v_1^2 = 1/2 (m +m_b) v2^2 + 1/2 k x^2

(60 kg ) ×9.8 m/s^2 ×17.1 m + 1/2 (60 kg ) (8 m/s)^2 = 1/2 (m +m_b)(0)^2 +1/2(2000 N/m) x^2

x = 3.5 m

Thus, the compression in the spring is 3.5 m

answer: at 24 february, 2014 the population gets twice of its initial population.

explanation:

since we have given that

the population for alpha city, t years after january 1, 2004 is given as

first we find out the initial population, i.e. t=0,

so, our quadratic equation becomes,

according to question, we have said that if the population has twice its initial population, then it becomes

so, time taken to reach this above mentioned population is given by

now, we know that time can't be negative so, we take t=9.149 years.

hence, at 24 february, 2014 the population gets twice of its initial population.