Question

extended response
8.1 linear momentum, force, and impulse

1. can a lighter object have more momentum than a heavier one? how?

a. no, because momentum is independent of the velocity of the object.
b. no, because momentum is independent of the mass of the object.
c. yes, if the lighter objects velocity is considerably high.
d. yes, if the lighter objects velocity is considerably low.

1. why does it hurt less when you fall on a softer surface?

a. the softer surface increases the duration of the impact, thereby reducing the effect of the force.
b. the softer surface decreases the duration of the impact, thereby reducing the effect of the force.
c. the softer surface increases the duration of the impact, thereby increasing the effect of the force.
d. the softer surface decreases the duration of the impact, thereby increasing the effect of the force.

1. can we use the equation ( f_{\text {net }}=\frac{delta p}{delta t} ) when the mass is constant?

a. no, because the given equation is applicable for the variable mass only.
b. no, because the given equation is not applicable for the constant mass.
c. yes, and the resultant equation is ( f=m v )
d. yes, and the resultant equation is ( f=m a )

Explanation:

44.

Step1: Recall the momentum formula

Momentum \(p = mv\), where \(m\) is mass and \(v\) is velocity.

Step2: Analyze each option

• Option a: Momentum \(p = mv\) is not independent of velocity. So, a is wrong.
• Option b: Momentum \(p = mv\) is not independent of mass. So, b is wrong.
• Option c: If \(m_1m_2v_2\). For example, \(m_1 = 1\space kg\), \(v_1= 100\space m/s\) and \(m_2 = 50\space kg\), \(v_2 = 1\space m/s\), then \(p_1=1\times100 = 100\space kg\cdot m/s\) and \(p_2=50\times1=50\space kg\cdot m/s\). So, c is correct.
• Option d: If velocity is low, \(p = mv\) will be small. So, d is wrong.

45.

Step1: Recall the impulse - momentum theorem

\(J=\Delta p=F_{avg}\Delta t\). When \(\Delta p\) (change in momentum) is constant (same fall, same initial and final velocity change).

Step2: Analyze each option

• Option a: Softer surface increases \(\Delta t\). From \(F_{avg}=\frac{\Delta p}{\Delta t}\), when \(\Delta t\) increases, \(F_{avg}\) (average force) decreases. So, a is correct.
• Option b: Softer surface does not decrease \(\Delta t\). So, b is wrong.
• Option c: Increasing \(\Delta t\) decreases \(F_{avg}\). So, c is wrong.
• Option d: Softer surface does not decrease \(\Delta t\). So, d is wrong.

46.

Step1: Expand the momentum formula for constant mass

If \(m\) is constant, \(\Delta p=m\Delta v\). Then \(F_{net}=\frac{\Delta p}{\Delta t}=\frac{m\Delta v}{\Delta t}\). Since \(a = \frac{\Delta v}{\Delta t}\), we get \(F_{net}=ma\)

Step2: Analyze each option

• Option a: The formula \(F_{net}=\frac{\Delta p}{\Delta t}\) is applicable for both constant and variable mass. For constant mass \(F = ma\). So, a is wrong.
• Option b: The formula is applicable for constant mass (as shown above). So, b is wrong.
• Option c: \(F = mv\) is momentum formula. \(F = ma\) is the correct resultant. So, c is wrong.
• Option d: As shown in Step1, when \(m\) is constant \(F_{net}=\frac{\Delta p}{\Delta t}=ma\). So, d is correct.

Answer:

1. C. Yes, if the lighter object's velocity is considerably high.
2. A. The softer surface increases the duration of the impact, thereby reducing the effect of the force.
3. D. Yes, and the resultant equation is \(F = ma\)