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Physics / Chapter 9: Center of Mass and Linear Momentum / Section 9-6

Conservation of Linear Momentum — Question 7

Section 9-6 · Physics pages 1264–1265

Question 7 Sequential throws with distinct launch references A cart and two 1 kg packages are initially at rest on a frictionless track. The empty cart has mass 8 kg. Both packages are thrown rightward, one at a time, each at 5 m/s relative to the cart immediately after that throw. cart + package 1 kg first 8 kg cart 1 kg second Each throw uses the cart velocity immediately after that throw; the first released package is then outside the remaining system. (a) Find cart velocity after each throw. (b) Compare with throwing both packages together at 5 m/s relative to the final cart.
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Question 7 - Solution Sequential throws with distinct launch references cart + package 1 kg first 8 kg cart 1 kg second (a) Derivation and calculation For the first throw, the remaining cart-package mass is 9 kg: 9V1 + (V1 + 5) = 0 ⇒V1 = −0.50 m/s . The remaining subsystem begins the second throw with momentum 9V1 = −4.5: 8V2 + (V2 + 5) = −4.5 ⇒V2 = −19 18 = −1.056 m/s . (b) Result and interpretation A simultaneous throw instead satisfies 8V + 2(V + 5) = 0 ⇒V = −1.0 m/s . Sequential throws produce slightly greater recoil because the second relative-speed condition uses a different moving launch frame. The first package is not accelerated again. Physical check The final momenta sum to zero: 8(−19/18) + (71/18) + 4.5 = 0.

Question 7

Sequential throws with distinct launch references

A cart and two 1kg1\,\mathrm{kg} packages are initially at rest on a frictionless track. The empty cart has mass 8kg8\,\mathrm{kg}. Both packages are thrown rightward, one at a time, each at 5m/s5\,\mathrm{m/s} relative to the cart immediately after that throw.

Diagram shown in the original SVG worksheet.

  1. Find cart velocity after each throw.

  2. Compare with throwing both packages together at 5m/s5\,\mathrm{m/s} relative to the final cart.

Question 7 - Solution

Sequential throws with distinct launch references

Diagram shown in the original SVG worksheet.

(a) Derivation and calculation

For the first throw, the remaining cart-package mass is 9 kg: 9V1+(V1+5)=0⇒V1=−0.50m/s.9V_1+(V_1+5)=0\Rightarrow\boxed{V_1=-0.50\ \mathrm{m/s}}. The remaining subsystem begins the second throw with momentum 9V1=−4.59V_1=-4.5: 8V2+(V2+5)=−4.5⇒V2=−1918=−1.056m/s.8V_2+(V_2+5)=-4.5\Rightarrow\boxed{V_2=-\frac{19}{18}=-1.056\ \mathrm{m/s}}.

(b) Result and interpretation

A simultaneous throw instead satisfies 8V+2(V+5)=0⇒V=−1.0m/s.8V+2(V+5)=0\Rightarrow\boxed{V=-1.0\ \mathrm{m/s}}. Sequential throws produce slightly greater recoil because the second relative-speed condition uses a different moving launch frame. The first package is not accelerated again.

Physical check

The final momenta sum to zero: 8(−19/18)+(71/18)+4.5=08(-19/18)+(71/18)+4.5=0.

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