Q: A brick is dropped from the roof of a building and strikes the ground in $$1.79\,\mathrm{s}. $$Neglecting air resistance, what is the height of the building?

$$h = \frac{1}{2} g t^2$ = \frac{1}{2} \times 9.8 \times (1.79)^2$ = 15.7\,\mathrm{m}$$

Q: A swimmer dives off a cliff with a horizontal leap of $$1.75\,\mathrm{m}$$ and falls $$9.00\,\mathrm{m}$$ vertically. Neglecting air resistance, what is the time taken to reach the water?

$$t = \sqrt{\frac{2(y-y_0$)}{g}} = \sqrt{\frac{2 \times 9.00}{9.8}} \approx 1.36\,\mathrm{s}$$

Q: A projectile is launched with initial velocity $$v_0 at an $$angle $$\theta$$ above the horizontal. What is the maximum height reached by the projectile?

$$h_{max} = \frac{v_0$^2 \sin^2$ \theta}{2g}$$

Q: A squirrel runs in a park with position given by $$\vec{r}(t) = (0.24t)\hat{i} + (0.36t^2$)\hat{j}. $$What is the magnitude of its velocity at $$t = 2.0\,\mathrm{s}$$?

$$v = \sqrt{(0.24)^2$ + (2 \times 0.36 \times 2.0)^2$} = \sqrt{0.0576 + 2.0736} \approx 1.47\,\mathrm{m/s}$$

Question 1

A car accelerates from rest with a constant acceleration $$a$$ along a straight road. Which equation correctly gives its position $$x as a $$function of time $$t$$?

Accepted Answer

$$x = x_0$ + v_0$ t + \frac{1}{2} a t^2$

Question 2

A brick is dropped from the roof of a building and strikes the ground in $$1.79\,\mathrm{s}. $$Neglecting air resistance, what is the height of the building?

Accepted Answer

$$h = \frac{1}{2} g t^2$ = \frac{1}{2} 	imes 9.8 	imes (1.79)^2$ = 15.7\,\mathrm{m}$$

Question 3

Which of the following best describes the velocity-time graph for an object moving with constant acceleration?

Accepted Answer

A straight line with a non-zero slope

Question 4

A swimmer dives off a cliff with a horizontal leap of $$1.75\,\mathrm{m}$$ and falls $$9.00\,\mathrm{m}$$ vertically. Neglecting air resistance, what is the time taken to reach the water?

Accepted Answer

$$t = \sqrt{\frac{2(y-y_0$)}{g}} = \sqrt{\frac{2 	imes 9.00}{9.8}} \approx 1.36\,\mathrm{s}$$

Question 5

A train moves at $$1.00\,\mathrm{m/s}$$ relative to the ground. A passenger walks at $$1.00\,\mathrm{m/s}$$ relative to the train in the same direction. What is the passenger's velocity relative to the ground?

Accepted Answer

$$v_{pg} = v_{pt} + v_{tg} = 1.00 + 1.00 = 2.00\,\mathrm{m/s}$$

Question 6

A projectile is launched with initial velocity $$v_0 at an $$angle $$	heta$$ above the horizontal. What is the maximum height reached by the projectile?

Accepted Answer

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

Question 7

Which of the following is the correct expression for centripetal acceleration $$a_{cent} of a $$particle moving in a circle of radius $$r$$ with speed $$v$$?

Accepted Answer

$$a_{cent} = \frac{v^2$}{r}$$

Question 8

A squirrel runs in a park with position given by $$\vec{r}(t) = (0.24t)\hat{i} + (0.36t^2$)\hat{j}. $$What is the magnitude of its velocity at $$t = 2.0\,\mathrm{s}$$?

Accepted Answer

$$v = \sqrt{(0.24)^2$ + (2 	imes 0.36 	imes 2.0)^2$} = \sqrt{0.0576 + 2.0736} \approx 1.47\,\mathrm{m/s}$$

Question 9

Which of the following statements about projectile motion is correct?

Accepted Answer

The horizontal velocity remains constant throughout the flight (neglecting air resistance)

Question 10

A cyclist rides at $$5\,\mathrm{m/s}$$ relative to the ground. A train moves at $$10\,\mathrm{m/s} in $$the same direction. What is the velocity of the cyclist relative to the train?

Accepted Answer

$$v_{ct} = v_{cg} - v_{tg} = 5 - 10 = -5\,\mathrm{m/s}$$

Question 11

An object moves with acceleration $$a(t) = -0.032(15-t)$$ and initial velocity $$v_0 = 8.0$$\,\mathrm{m/s}$$ at t=0$. What is the velocity at t=10$$\,\mathrm{s}$$?

Accepted Answer

$$v(10) = v_0$ + \$$int_0$$^{10} a(t) dt = 8.0 + [-0.032(15t - \frac{t^2$}{2})]_{0}^{10} = 8.0 + [-0.032(150 - 50)] = 8.0 + [-0.032 	imes 100] = 8.0 - 3.2 = 4.8\,\mathrm{m/s}$$

Question 12

Which of the following is the correct formula for the range $$R of a $$projectile launched at speed $$v_0$$ and angle $$	heta$$ above the horizontal?

Accepted Answer

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

Question 13

A particle moves in a circle of radius $$r$$ with period $$T. $$What is its speed $$v$$?

Accepted Answer

$$v = \frac{2\pi r}{T}$$

Question 14

A car traveling at $$110\,\mathrm{km/h}$$ must stop as quickly as possible. If the car slows down to $$50\,\mathrm{km/h} in$$ $$5\,\mathrm{s}$$, what is the average acceleration?

Accepted Answer

$$a = \frac{v_2$ - v_1$}{t} = \frac{(50 	imes 0.278) - (110 	imes 0.278)}{5} = \frac{13.9 - 30.58}{5} = -3.34\,\mathrm{m/s^2$}$$

Question 15

Which of the following is the correct expression for the instantaneous velocity $$v in $$terms of position $$x$$ and time $$t$$?

Accepted Answer

$$v = \frac{dx}{dt}$$

Question 16

A particle moves in three dimensions with position vector $$\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}. $$What is the magnitude of its velocity?

Accepted Answer

$$|\vec{v}| = \sqrt{\left(\frac{dx}{dt}\$$right)^2$$ + \left(\frac{dy}{dt}\$$right)^2$$ + \left(\frac{dz}{dt}\$$right)^2$$}$$

Question 17

Which of the following is the correct formula for the displacement $$\Delta x of an $$object under non-constant acceleration $$a(t)$$ over time interval $$t_0 to$$ $$t_1$$?

Accepted Answer

$$\Delta x = \int_{t_0$}^{t_1$} v(t) dt$$

DR1 – Motion in One and Two Dimensions

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