Q: If the truck now moves down an incline at angle $$\alpha to $$the horizontal, still accelerating with magnitude $$a$$, what is the new angle $$\varphi$$ the rope makes with the vertical?

$$\tan \varphi = \frac{a + g \sin \alpha}{g \cos \alpha}$$

Q: A projectile is launched from height $$h$$ with initial velocity $$v_0 at $$angle $$\theta$$ above the horizontal. What is the time $$t_{max} to $$reach the highest point?

$$t_{max} = \frac{v_0$ \sin \theta}{g}$$

Question 1

A truck accelerates to the right with acceleration $$a on $$level ground. A bucket of mass $$M is $$attached to the truck by a massless rope, which settles at an angle $$	heta$$ with respect to the vertical. What is the correct expression for $$	an 	heta$$?

Accepted Answer

$$	an 	heta = \frac{a}{g}$$

Question 2

For the same truck and bucket system, what is the tension $$T_1 in $$the rope once the bucket settles at angle $$	heta$$?

Accepted Answer

$$T_1 = M$$ \sqrt{$$a^2$$ + $$g^2$$}$$

Question 3

If the truck now moves down an incline at angle $$\alpha to $$the horizontal, still accelerating with magnitude $$a$$, what is the new angle $$\varphi$$ the rope makes with the vertical?

Accepted Answer

$$	an \varphi = \frac{a + g \sin \alpha}{g \cos \alpha}$$

Question 4

For the inclined truck scenario, what is the new tension $$T_2 in $$the rope?

Accepted Answer

$$T_2 = M$$ \sqrt{(a + g \sin \$$alpha)^2 + (g$$ \cos \$$alpha)^2$$}$$

Question 5

A projectile is launched from $$(0, h)$$ with initial velocity components $$v_{0x}$$ and $$v_{0y}. $$What is the $$y-$$coordinate as a function of $$x$$?

Accepted Answer

$$y(x) = -\frac{g}{2 v_{0x}^2} x^2$ + \frac{v_{0y}}{v_{0x}} x + h$$

Question 6

For the projectile above, what is the horizontal range $$x_f$$ where the projectile lands ($$y_f = 0$$)?

Accepted Answer

$$x_f = \frac{v_{0x}}{g} \left( v_{0y} + \sqrt{v_{0y}^2 + 2 g h} \right)$$

Question 7

In an Atwood machine with masses $$M_1$$ and $$M_2$$ ($$M_2$$ \u003e $$M_1$$), what is the acceleration $$a of $$mass $$M_1$$?

Accepted Answer

$$a = \frac{M_2$ - M_1$}{M_1$ + M_2$} g$$

Question 8

For the same Atwood machine, what is the tension $$T in $$the rope?

Accepted Answer

$$T = \frac{2 M_1$ M_2$}{M_1$ + M_2$} g$$

Question 9

If the Atwood machine's pulley is accelerated upward with acceleration $$a_p$$, what is the new tension in the rope?

Accepted Answer

$$T = \frac{2 M_1$ M_2$}{M_1$ + M_2$} (g + a_p)$$

Question 10

A particle moves such that its velocity is given by $$v = c x^2$, where c$ is a constant. If x(0$$) = $$x_0$$, what is the position $$x(t) as a $$function of time?

Accepted Answer

$$x(t) = \frac{x_0$}{1 - c x_0$ t}$$

Question 11

For the particle in the previous question, what happens to $$x(t) as$$ $$t$$ approaches $$1/(c x_0$)$$?

Accepted Answer

$$x(t)$$ approaches infinity

Question 12

A projectile is launched from height $$h$$ with initial velocity $$v_0 at $$angle $$	heta$$ above the horizontal. What is the time $$t_{max} to $$reach the highest point?

Accepted Answer

$$t_{max} = \frac{v_0$ \sin 	heta}{g}$$

Question 13

In the Atwood machine, if $$M_1$$ = $$M_2$$, what is the acceleration of the system?

Accepted Answer

$$0$$

Question 14

A projectile is launched from the ground with $$v_0 at $$angle $$	heta. $$What is the maximum height $$H$$ reached?

Accepted Answer

$$H = \frac{v_0$^2 \sin^2$ 	heta}{2g}$$

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