Question

A shopper is pushing their trolley westwards across an aisle at a constant speed of 1.2 ms-1. Another shopper with groceries in their trolley is moving north out of the aisle at 1.6 ms-1 and the trollies collide. The trollies jam together and end up moving at a velocity of 1.234 ms-1 in a direction 25.7° W of N. If 38 J of energy is lost into noise, deforming the trolleys etc. during the collision, what is the mass of a trolley, (assuming they are the same type of trolley) and what was the mass of the second shopper’s groceries?

Answer 1

Step 1:

this is a completely inelastic collision, So

Using momentum conservation in x-direction:

Pix = Pfx

m1*V1x + m2*V2x = (m1 + m2)*Vx

m1 = mass of unladen trolley

m2 = mass of trolley with groceries

V1x = Initial speed of m1 in x-direction = 1.2 m/sec

V2x = Initial speed of m2 in x-direction = 0 m/sec

V = final speed of trollies = 1.234 m/sec at 25.7 deg W of N = 1.234 m/sec at 64.3 deg N of W

(assuming west is +x direction and North in +ve y-direction)

Vx = final speed of trollies in x-direction = 1.234*cos 64.3 deg = 0.535 m/sec

So,

m1*1.2 + m2*0 = (m1 + m2)*0.535

m1*1.2 = (m1 + m2)*0.535

Using momentum conservation in y-direction:

Piy = Pfy

m1*V1y + m2*V2y = (m1 + m2)*Vy

m1 = mass of unladen trolley

m2 = mass of trolley with groceries

V1y = Initial speed of m1 in y-direction = 0 m/sec

V2y = Initial speed of m2 in y-direction = 1.6 m/sec

V = final speed of trollies = 1.234 m/sec at 25.7 deg W of N = 1.234 m/sec at 64.3 deg N of W

(assuming west is +x direction and North in +ve y-direction)

Vy = final speed of trollies in y-direction = 1.234*sin 64.3 deg = 1.112 m/sec

So,

m1*0 + m2*1.6 = (m1 + m2)*1.112

m2*1.6 = (m1 + m2)*1.112

Now given that kinetic energy lost is 38 J, So

dKE = KEf - KEi = -38 J

KEi = KEf + 38 J

(1/2)*m1*V1^2 + (1/2)*m2*V2^2 = (1/2)*(m1 + m2)*V^2 + 38

(1/2)*m1*1.2^2 + (1/2)*m2*1.6^2 = (1/2)*(m1 + m2)*1.234^2 + 2*38

m1*1.2^2 + m2*1.6^2 = (m1 + m2)*1.234^2 + 2*38

m1*1.2 = (m1 + m2)*0.535

m2*1.6 = (m1 + m2)*1.112

divide last two equation

m1/m2 = 1.6*0.535/(1.2*1.112) = 0.6415

Now divide 1st equation by m2

m1*1.2^2 + m2*1.6^2 = (m1 + m2)*1.234^2 + 2*38

m1*1.2^2/m2 + m2*1.6^2/m2 = (m1/m2 + m2/m2)*1.234^2 + 2*38/m2

(m1/m2)*1.2^2 + 1.6^2 = (m1/m2 + 1)*1.234^2 + 2*38/m2

Since m1/m2 = 0.6415, So

0.6415*1.2^2 + 1.6^2 = (0.6415 + 1)*1.234^2 + 2*38/m2

2*38/m2 = (0.6415*1.2^2 + 1.6^2 - (0.6415 + 1)*1.234^2)

2*38/m2 = 0.98416

m2 = 2*38/0.98416

m2 = 77.22 kg

So, m1 = m2*0.6415

m1 = 77.22*0.6415 = 49.54 kg

mass of unladen trolley = 49.54 kg

mass of trolley filled with groceries = 77.22 kg

mass of groceries = m2 - m1 = 77.22 - 49.54 = 27.68 kg

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