Q: Given the block in the previous question is launched at $$v = 8.97\ \mathrm{m/s}$$ from a $$45^\circ$$ ramp at a height $$H = 1\ \mathrm{m}$$ above the ground, what is the correct quadratic equation to solve for the range $$R of $$the block's flight?

$$0 = -\frac{g}{2v^2$\cos^2$\theta}R^2$ + (\tan\theta)R + H$$

Q: For the post-impact swing of the two stuck-together balls, what is the maximum height $$h'$$ they reach, in terms of $$L$$ and $$\theta$$?

$$h' = \frac{1}{4}L(1 - \cos\theta)$$

Q: A block is launched from a spring and travels off a ramp at an angle $$\theta$$ with speed $$v_0. $$What is the time $$T it $$spends in the air before landing, in terms of $$R ($$range), $$v_0$$, and $$\theta$$?

$$T = \frac{R}{v_0$\cos\theta}$$

Q: A block is launched off a ramp at $$45^\circ$$ with speed $$v_0$$ from a height $$H$$ above the ground. Which equation correctly expresses the vertical position $$y as a $$function of time $$t$$?

$$y = H + v_0$\sin\theta\ t - \frac{1}{2}gt^2$

Question 1

A block of mass $$m = 0.1\ \mathrm{kg} is $$compressed against a spring with spring constant $$k = 1000\ \mathrm{N/m} by$$ $$x = 0.10\ \mathrm{m}$$ and released on a frictionless surface, then launched off a $$45^\circ$$ ramp. What is the launch speed $$v of $$the block as it leaves the ramp?

Accepted Answer

$$v = \sqrt{\frac{2kx^2$}{m}}$$

Question 2

Given the block in the previous question is launched at $$v = 8.97\ \mathrm{m/s}$$ from a $$45^\circ$$ ramp at a height $$H = 1\ \mathrm{m}$$ above the ground, what is the correct quadratic equation to solve for the range $$R of $$the block's flight?

Accepted Answer

$$0 = -\frac{g}{2v^2$\cos^2$	heta}R^2$ + (	an	heta)R + H$$

Question 3

Two balls of identical mass $$m_1$$ = $$m_2 = 45$$\ \mathrm{g}$$ are suspended from a string of length L = 30$$\ \mathrm{cm}$$. Ball A$ is pulled out to a 66$$^\circ$$ angle and released, colliding and sticking to ball B$. What is the height h$ that ball A$ falls before impact, in terms of L$ and $$	heta$$?

Accepted Answer

$$h = L(1 - \cos	heta)$$

Question 4

After the collision in the previous question, what is the speed $$v' of $$the combined mass immediately after impact?

Accepted Answer

$$v' = \frac{v}{2}$$

Question 5

For the post-impact swing of the two stuck-together balls, what is the maximum height $$h'$$ they reach, in terms of $$L$$ and $$	heta$$?

Accepted Answer

$$h' = \frac{1}{4}L(1 - \cos	heta)$$

Question 6

A ball slides down three different frictionless tracks (A, B, C), all starting at the same height. Which track will result in the highest final speed of the ball at the end?

Accepted Answer

All tracks result in the same final speed

Question 7

Why do all three tracks in the previous question result in the same final speed for the ball?

Accepted Answer

Because mechanical energy is conserved and only the change in height matters

Question 8

Three balls are thrown off a tall building with the same initial speed but in different directions: Ball A vertically, Ball B horizontally, Ball C at $$45^\circ$$ above the horizontal. Which ball has the greatest final speed just before hitting the ground?

Accepted Answer

All three balls have the same final speed

Question 9

For the three balls thrown from the building, what principle explains why their final speeds are equal?

Accepted Answer

Conservation of mechanical energy

Question 10

A block is launched from a spring and travels off a ramp at an angle $$	heta$$ with speed $$v_0. $$What is the time $$T it $$spends in the air before landing, in terms of $$R ($$range), $$v_0$$, and $$	heta$$?

Accepted Answer

$$T = \frac{R}{v_0$\cos	heta}$$

Question 11

A block is launched off a ramp at $$45^\circ$$ with speed $$v_0$$ from a height $$H$$ above the ground. Which equation correctly expresses the vertical position $$y as a $$function of time $$t$$?

Accepted Answer

$$y = H + v_0$\sin	heta\ t - \frac{1}{2}gt^2$

Question 12

A block is launched off a ramp at $$45^\circ$$ with speed $$v_0$$ from a height $$H$$ above the ground. Which equation correctly expresses the horizontal position $$x as a $$function of time $$t$$?

Accepted Answer

$$x = v_0$\cos	heta\ t$$

Question 13

If a ball is launched horizontally from a height $$H$$ with speed $$v_0$$, what is its final speed $$v_f$$ just before hitting the ground?

Accepted Answer

$$v_f = \sqrt{v_0$^2 + 2gH}$$

Question 14

A ball is launched at $$45^\circ$$ above the horizontal from a height $$H$$ with speed $$v_0. $$What is its final speed $$v_f$$ just before hitting the ground?

Accepted Answer

$$v_f = \sqrt{v_0$^2 + 2gH}$$

Question 15

A ball is dropped vertically from a height $$H$$ with zero initial speed. What is its final speed $$v_f$$ just before hitting the ground?

Accepted Answer

$$v_f = \sqrt{2gH}$$

Question 16

In the spring-block-ramp scenario, why is mechanical energy conserved as the block slides along the ramp?

Accepted Answer

Because the ramp is frictionless

Question 17

In the two-ball pendulum collision, why is momentum conserved during the impact?

Accepted Answer

Because the impact is isolated and there is no external force

Question 18

A ball slides down a frictionless ramp from height $$h$$ and reaches speed $$v at $$the bottom. Which equation relates $$v$$ and $$h$$?

Accepted Answer

$$v = \sqrt{2gh}$$

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