Q: You are pushing a 50kg box up a 20ft long ramp that rises 3ft to the back of a truck. Assuming no friction, what is the average force required to move the box up the ramp at constant speed?

$$F = mg \sin \theta = 50 \times 9.8 \times \sin(\arcsin(3/20))$$

Q: If you want to double the force required to push the 50kg box up the ramp, what should the new angle $$\theta of $$the ramp be (relative to the horizontal)?

$$\theta = \arcsin(2 \sin(\arcsin(3/20)))$$

Q: In the scenario where two cars collide perpendicularly (one at 45mph, 4000lbs; the other at 25mph, 3750lbs), what is the direction of the combined wreck's velocity relative to the original direction of the 4000lb car?

$$\theta = \arctan\left(\frac{m_2$ v_2$}{m_1$ v_1$}\right)$$

Q: If the ramp in the moving truck scenario is made steeper while keeping its length constant, what happens to the force required to push the box up at constant speed?

The required force increases because $$\sin \theta$$ increases.

Question 1

A 24g mass rests on a horizontal surface and is connected by a string over a frictionless pulley to a 22g mass hanging vertically. When the 22g mass is released, it begins to fall. If the system accelerates at less than $$g$$, what is the coefficient of friction $$\mu$$ between the 24g mass and the surface?

Accepted Answer

$$\mu = \frac{m_2$ g - (m_1$ + m_2$)a}{m_1$ g}$$

Question 2

You are pushing a 50kg box up a 20ft long ramp that rises 3ft to the back of a truck. Assuming no friction, what is the average force required to move the box up the ramp at constant speed?

Accepted Answer

$$F = mg \sin 	heta = 50 	imes 9.8 	imes \sin(\arcsin(3/20))$$

Question 3

If you want to double the force required to push the 50kg box up the ramp, what should the new angle $$	heta of $$the ramp be (relative to the horizontal)?

Accepted Answer

$$	heta = \arcsin(2 \sin(\arcsin(3/20)))$$

Question 4

A circus acrobat of unknown mass falls from a height of 9m onto a square net held by 25 identical springs per side, each with $$k = 10\,\mathrm{N/m}. If $$the net depresses 3.9m, what is the mass of the acrobat?

Accepted Answer

$$m = \frac{N k x^2$}{2 g h}$$, where $$N = 100$$, $$x = 3.9$$, $$h = 9$$

Question 5

If the netting in the circus act is depressed to 4.9m instead of 3.9m, how far would the acrobat have fallen before hitting the net, assuming the same mass and spring constants?

Accepted Answer

$$h = \frac{N k x^2$}{2 m g}$$, where $$x = 4.9$$

Question 6

A car of mass 4000lbs ($$1\,\mathrm{lb} = 0.4536\,\mathrm{kg}) is $$moving at 45mph and is struck perpendicularly by a 3750lb car moving at 25mph. What is the magnitude of the velocity of the combined wreck immediately after the collision, assuming a perfectly inelastic collision?

Accepted Answer

$$v = \sqrt{\left(\frac{m_1$ v_1$}{m_1$ + m_2$}\$$right)^2$$ + \left(\frac{m_2$ v_2$}{m_1$ + m_2$}\$$right)^2$$}$$

Question 7

In the scenario where two cars collide perpendicularly (one at 45mph, 4000lbs; the other at 25mph, 3750lbs), what is the direction of the combined wreck's velocity relative to the original direction of the 4000lb car?

Accepted Answer

$$	heta = \arctan\left(\frac{m_2$ v_2$}{m_1$ v_1$}ight)$$

Question 8

If the two cars (4000lbs at 45mph and 3750lbs at 25mph) were instead traveling in the same direction and the 3750lb car rear-ends the 4000lb car at 65mph, what is the final velocity of the combined mass after a perfectly inelastic collision?

Accepted Answer

$$v_f = \frac{m_1$ v_1$ + m_2$ v_2$}{m_1$ + m_2$}$$

Question 9

When pushing a box up a ramp at constant speed, which of the following best describes the relationship between the force applied parallel to the ramp $$F$$, the mass $$m$$, gravity $$g$$, and the angle $$	heta of $$the ramp?

Accepted Answer

$$F = mg \sin 	heta$$

Question 10

If the ramp in the moving truck scenario is made steeper while keeping its length constant, what happens to the force required to push the box up at constant speed?

Accepted Answer

The required force increases because $$\sin 	heta$$ increases.

Question 11

In the circus net scenario, if the acrobat's mass is doubled, how does the maximum depression $$x of $$the net change, assuming the same fall height and spring constants?

Accepted Answer

$$x$$ increases by a factor of $$\sqrt{2}$$

Question 12

Which of the following best describes the energy transformation when the acrobat falls onto the net and the net is depressed?

Accepted Answer

Gravitational potential energy is converted into elastic potential energy.

Question 13

If the coefficient of friction between the box and the ramp is $$\mu$$, what is the minimum force required to push the box up the ramp at constant speed?

Accepted Answer

$$F = mg (\sin 	heta + \mu \cos 	heta)$$

Question 14

In the car collision scenario, why is it necessary to convert the masses from pounds to kilograms and velocities from mph to m/s before calculating momentum?

Accepted Answer

Because SI units are required for consistency in the momentum calculation.

Question 15

If the net in the circus act is replaced with one having springs of $$k = 20\,\mathrm{N/m} ($$double the original), what happens to the maximum depression $$x$$ for the same acrobat and fall height?

Accepted Answer

$$x$$ decreases by a factor of $$1/\sqrt{2}$$

Physics Problem-Solving Guidance: Forces, Friction, Inclined Planes, Springs, and Collisions

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