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Two cars collide at an icy intersection and stick together afterward. The first car has a mass of 1800 kg and was approaching at 5.00 m/s due south. The second car has a mass of 700 kg and was approaching at 21.0 m/s due west. (a) Calculate the final velocity of the cars. (b) How much kinetic energy is lost in the collision? (This energy goes into deformation of the cars.)

User BanikPyco
by
7.7k points

1 Answer

6 votes

Answer:

The final velocity of the cars is 6.894 m/s

The kinetic energy lost is 117,441 J

Step-by-step explanation:

Using the conservation of the linear momentum :


P_i = P_f

Where
P_i is the inicial linear momentum and
P_f is the final linear momemtum.

The linear momentum is calculated by:


P = MV

where M is the mass and V is the velocity.

First we identify the directions of the velocity of both cars:

  • the first car is moving in the axis y for the south direction, we will take this direction like positive.
  • the second car is moving in the axis x to the west, we will take this direction like positive.

The conservation of the linear momentum is made on direction x, so:


P_i = P_f\\M_(1)V_(1ix) +M_(2)V_(2ix) = V_(sx)(M_1+M_2)

where
M_1 is the mass of the car 1,
V_(1ix) is the car 1's inicial velocity in the axis x,
M_2 is the mass of the car 2,
V_(2ix) is the car 2's inicial velocity in the axis x and
V_(sx) is the velocity of the car 1 and car 2 after the collition.

The car 1 just move in the axis y so, it dont have horizontal velocity (
V_(1ix) = 0)

now the equation is:


M_2V_(2ix) = V_(sx)(M_1 +M_2)

Replacing the values, we get:

(700 kg)(21 m/s) =
V_(sx)( 1800 kg + 700 kg)

solving for
V_(sx):


V_(sx) = (700(21))/(1800+700)


V_(sx) = 5.88 m/s

Now we do the conservation of the linear momentum on direction y:


P_i = P_f\\M_(1)V_(1iy) +M_(2)V_(2iy) = V_(sy)(M_1+M_2)

where
M_1 is the mass of the car 1,
V_(1iy) is the car 1's inicial velocity in the axis y,
M_2 is the mass of the car 2,
V_(2iy) is the car 2's inicial velocity in the axis y and
V_(sy) is the velocity of the car 1 and car 2 together after the collition.

The car 2 just move in the axis x so, it don't have horizontal velocity (
V_(2iy) = 0)

now the equation is:


M_1V_(1iy) = V_(sy)(M_1 +M_2)

Replacing the values, we get:


1800(5 m/s) = V_(sy)(1800 +700)

solving for
V_(sx):


V_(sy) = (1800(5))/(1800+700)


V_(sy) = 3.6 m/s

Now, we have the two components of the velocity and using pythagorean theorem we find the answer as:


V_f = √(5.88^2+3.6^2)


V_f = 6.894 m/s

Finally we have to find the kinetic energy lost, so the kinetic energy is calculated by:

K =
(1)/(2)MV^2

so, ΔK =
K_f -K_i

where
K_f is the final kinetic energy and
K_i is the inicial kinetic energy.

then:


K_i = (1)/(2)(1800)(5)^2+(1)/(2)(700)(21)^2= 176,850 J


K_f = (1)/(2)(1800+700)(6.894)^2 = 59,409 J

so:

ΔK = 59409 - 176850 = -117441 J

User Terafor
by
7.9k points
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