Question

Two carts sit on a horizontal, frictionless track; the spring between them is compressed. The small cart has mass m, and the mass of the larger cart is M = 5.29m. NOTE: Every velocity needs magnitude and direction (given by the sign).

a) Suppose the carts are initially at rest, and after the "explosion" the smaller cart is moving at velocity v = +4.88 m/s. - Find the velocity of the larger cart. V =

Assume now that the mass of the smaller cart is m = 8.91 kg. Assuming there is no loss of energy: find the energy stored in the spring before the explosion. Wk =

If the spring has spring constant k = 935 N/m: find x, the distance the spring was compressed before the "explosion".

b) Suppose the carts are initially moving together, with the spring compressed between them, at constant velocity vo = +9.39 m/s. After the "explosion", the smaller cart is moving at velocity v = +4.88 m/s. Find the velocity of the larger cart.

c) Suppose now that the small cart (mass m) is initially moving at velocity vo = +3.3 m/s. At what velocity would the large cart (mass 5.29m) have to be moving so, when they collide and stick together, they remain at rest?

If you can show the work/ provide explanation I would greatly appreciate it :) Thanks

Answer 1

mass of small cart = m
mass of larger cart = 5.29 m

a) Suppose the carts are initially at rest, and after the "explosion" the smaller cart is moving at velocity +4.88 m/s.

Momentum is always conserved!
Initial momentum = 0, since both carts are at rest
Final momentum = (m * 4.88) + (5.29 * m * v)

Final momentum = Initial momentum
(m * 4.88) + (5.29 * m * v) = 0
(m * 4.88) = -(5.29 * m * v)
v = -0.922 m/s
The velocity of the larger cart = 0.922 m/s in the opposite direction of smaller car!


- If the mass of the smaller cart is 8.91 kg, find the energy supplied by the spring to the carts. J
The energy supplied by the spring is transferred to the carts.
Total KE of carts = energy supplied by the spring
KE = ½ * mass * velocity²
Mass of larger car = 5.29* 8.91 = 47.134 kg
Total KE = ½ * 8.91* (4.88)² + ½ * 47.134 * (0.922)²

The energy supplied by the spring = 126.124J
Spring potential energy = ½ * k * x²
½ * 935 * x² = 126.124
distance =0.52m

b) Suppose the carts are initially moving together at constant velocity +9.39 m/s. After the "explosion", the smaller cart is moving at velocity +4.88 m/s. Find the velocity of the larger cart.
m/s

Total Initial momentum = (8.91+ 47.134) * 9.39=526.25

Total Final momentum = (8.91* 4.88) +(47.134* vf)

(8.91* 4.88) +(47.134* vf) = 8.91+ 47.134) * 9.39=526.25

vf = 10.24m/s

c)

initial momentum = m*3.3 +5.29m*v

final momentum =0

by momentum conservation

m*3.3 +5.29m*v =0

v = -0.624m/s