Flashcards: Projectile Motion Fundamentals — 79 cards

What type of motion is projectile motion?

Two-dimensional motion along independent horizontal and vertical axes.

What can be ignored when analyzing horizontal motion in projectile motion?

Vertical motion can be ignored.

What can be ignored when analyzing vertical motion in projectile motion?

Horizontal motion can be ignored.

When is air resistance considered in the standard projectile model?

Only if the problem explicitly includes it.

What value is typically used for gravitational acceleration in projectile problems?

9.8 m/s² or approximately 10 m/s² unless specified otherwise.

What is the direction and magnitude of acceleration for a particle under gravity alone?

Vertically downward with magnitude g.

What is the projectile acceleration vector with upward and rightward positive?

$\vec a = -g\hat j$

What is the horizontal acceleration of a projectile in standard conditions?

Zero.

Why does the horizontal velocity remain constant during projectile flight?

Because the horizontal acceleration is zero.

What are the initial velocity components of a particle projected at 53° to vertical at 100 m/s?

$u_x=80\,\mathrm{m\,s^{-1}}$ and $u_y=60\,\mathrm{m\,s^{-1}}$.

What is the velocity vector $\vec v(t)$ at time t for the example with $g=10\,\mathrm{m\,s^{-2}}$?

$\vec v(t)=80\hat i+(60-10t)\hat j\,\mathrm{m\,s^{-1}}$.

What is the position vector $\vec r(t)$ at time t for the particle projected from the origin?

$\vec r(t)=80t\hat i+(60t-5t^2)\hat j\,\mathrm{m}$.

How is the time to reach the highest point $t_{\mathrm{max}}$ calculated?

By setting vertical velocity to zero: $t_{\mathrm{max}}=\frac{u_y}{g}$.

What are the time of flight and range for the particle returning to the same level?

Time of flight $T=12\,\mathrm{s}$ and range $R=960\,\mathrm{m}$.

How is the maximum height $H_{\mathrm{max}}$ calculated in the example?

$H_{\mathrm{max}}=\frac{u_y^2}{2g}$ equals 180 m.

What is the trajectory equation after eliminating time from $x=80t$ and $y=60t-5t^2$?

$y=-\frac{x^2}{1280}+\frac{3x}{4}$, a downward-opening parabola.

What is the initial velocity vector of a projectile launched at speed u and angle θ?

$\vec{u}=u\cos\theta\,\hat{i}+u\sin\theta\,\hat{j}$

What is the velocity vector of a projectile at time t?

$\vec{v}=u\cos\theta\,\hat{i}+(u\sin\theta-gt)\,\hat{j}$

What are the coordinates of a projectile at time t?

$x=u\cos\theta\,t$ and $y=u\sin\theta\,t-\frac{1}{2}gt^2$

What is the trajectory equation of a projectile eliminating time t?

$y=x\tan\theta-\frac{gx^2}{2u^2\cos^2\theta}$

What is the time to reach maximum height for a projectile returning to launch level?

$t_{\mathrm{up}}=\frac{u\sin\theta}{g}$

What is the total time of flight for a projectile returning to launch level?

$T=\frac{2u\sin\theta}{g}$

What is the maximum height and range of a projectile returning to launch level?

Maximum height: $H=\frac{u^2\sin^2\theta}{2g}$; Range: $R=\frac{u^2\sin 2\theta}{g}$

When is velocity perpendicular to acceleration in vector terms?

When their dot product $\vec v\cdot\vec a=0$.

At what time is velocity perpendicular to acceleration for $\vec v=80\hat i+(60-10t)\hat j$ and $\vec a=-10\hat j$?

At $t=6\text{ s}$, the highest point.

When is velocity perpendicular to initial velocity mathematically?

When their dot product $\vec v\cdot\vec u=0$.

What is the mathematical time when $\vec u=80\hat i+60\hat j$ and $\vec v=80\hat i+(60-10t)\hat j$ are perpendicular?

At $t=\frac{50}{3}\text{ s}$, which is not physically possible if flight ends at 12 s.

What is the final velocity formula when a projectile lands at original height?

$\vec v_f=u\cos\theta\,\hat i - u\sin\theta\,\hat j$.

What is the formula for torque about a point?

$\vec{\tau} = \vec{r} \times \vec{F}$

What does differentiating a position vector once give?

Velocity

What does differentiating a position vector twice give?

Acceleration

How is force related to acceleration and mass?

Force equals mass times acceleration

How do you find the horizontal range from a trajectory equation?

Set $y=0$ and take the nonzero value of $x$.

What horizontal range results from setting $y=0$ in $y=x-\frac{x^2}{80}$?

$80\ \mathrm{m}$

How do you find the maximum height from a trajectory equation?

Set $\frac{dy}{dx}=0$ to find $x$, then substitute into the equation.

At what angle is range maximum for fixed initial speed in ground-to-ground motion?

45 degrees

What is the maximum range value for fixed initial speed in ground-to-ground motion?

$\frac{u^2}{g}$

Which launch angles produce the same range for fixed speed and equal launch and landing levels?

$\theta$ and $90^\circ - \theta$

How is projectile motion solved in terms of velocity components?

By resolving initial velocity into horizontal and vertical components and treating motions separately

How is the maximum height of a projectile expressed in terms of launch speed and angle?

$H=\frac{u^2\sin^2\theta}{2g}$

What formula gives the horizontal range for ground-to-ground projectile motion?

$R=\frac{u^2\sin 2\theta}{g}$

How are the initial velocity components $u_x$ and $u_y$ defined for projectile motion?

$u_x=u\cos\theta$ and $u_y=u\sin\theta$

What is the time of flight formula using initial vertical velocity component $u_y$?

$T=\frac{2u_y}{g}$

At which launch angle is the range of a projectile maximized for fixed speed?

At $\theta=45^\circ$

Why does the range maximize at $\theta=45^\circ$ for fixed launch speed?

Because $\sin 2\theta$ reaches its maximum value of 1 there

Why do complementary angles $\theta$ and $90^\circ-\theta$ yield the same projectile range?

Because $\sin(180^\circ-2\theta)=\sin 2\theta$

What vertical velocity components do two projectiles with equal max heights share?

They have equal vertical components $u_1\sin\theta_1 = u_2\sin\theta_2$.

On what does the range of two projectiles with equal max heights depend?

Their ranges depend on the products $u_x u_y$.

What is the launch angle $\theta$ if a projectile has the same horizontal range and max height?

The launch angle satisfies $\tan\theta = 4$.

How do you check if motion is one-dimensional or two-dimensional using velocity components?

Calculate $\tan\alpha = v_y / v_x$ and check if the direction angle remains constant.

What is the formula for the new projectile range on a lower level?

$R' = \frac{R}{2}+x$

How is the additional horizontal distance after landing calculated with changed gravity?

$x = u \cos \theta \times \frac{2h_{\max}}{g'}$

Which axes are chosen for projectile motion on an inclined plane?

Axes parallel and perpendicular to the plane.

How is gravitational acceleration resolved on an inclined plane at angle θ?

Into $g_{\perp}=g\cos\theta$ perpendicular and $g_{\parallel}=g\sin\theta$ along the plane.

What must be done to initial velocity u at angle θ before applying kinematics on an incline?

Resolve it into components along the chosen axes.

How is the position vector of a particle with coordinates x and y expressed?

$\vec r=x\hat i+y\hat j$

What does the course treat projectile motion as part of?

Kinematics involving motion along two independent axes.

What formula relates the angle $\theta$ of velocity with horizontal at time t?

$\tan\theta=\frac{60-10t}{80}$.

What is the formula for change in velocity between launch and landing at same height?

$\Delta\vec v = -2u\sin\theta\,\hat j$.

How is torque of gravitational force about origin expressed at position $\vec r=x\hat i + y\hat j$?

$\vec\tau = mgx\hat k$.

What is the angular momentum magnitude at highest point about launch point?

$L = mH u \cos\theta$.

How do you find velocity from force if the particle starts at rest?

Integrate force divided by mass

How do you find position vector from velocity?

Integrate velocity

How is velocity obtained from $\vec{r} = \frac{10}{3}t^3\hat{i} + 5t^2\hat{j}$?

By differentiating $\vec{r}$ with respect to time

How is velocity at $t=1\ \mathrm{s}$ found for $\vec{r} = \frac{10}{3}t^3\hat{i} + 5t^2\hat{j}$?

By substituting $t=1$ into the derivative of $\vec{r}$

At what $x$ and height does $y=x-\frac{x^2}{80}$ reach its maximum?

Maximum at $x=40\ \mathrm{m}$ and height $20\ \mathrm{m}$.

How do you identify $\tan\theta$ from a given trajectory equation?

From the coefficient of $x$ in the equation.

How do you identify initial speed $u$ from a trajectory equation?

From the coefficient of $x^2$ in the equation.

What happens to horizontal velocity when a horizontal force acts besides gravity?

Horizontal velocity is not constant

Can initial and final vertical speeds be related if no vertical force acts besides gravity?

Yes, through vertical motion equations

What are the horizontal and vertical components of velocity for a projectile launched at 50 m/s at 37°?

40 m/s horizontal and 30 m/s vertical

What formula expresses the range $R$ of a projectile with zero horizontal acceleration?

$R=\frac{2 u_x u_y}{g}$.

Why is the range greater for the projectile with the greater product $u_x u_y$?

Because range $R$ is proportional to $2 u_x u_y / g$ when horizontal acceleration is zero.

What type of motion results from zero initial velocity and constant acceleration?

One-dimensional straight-line motion.

How is velocity direction relative to the x-axis determined from position vector $\vec r(t)$?

Differentiate $\vec r(t)$ to get velocity, then use $\tan\alpha = v_y / v_x$ for direction.

What is the new-to-old range ratio when gravity changes to $g' = \frac{g}{81}$?

95

How is the inclined-plane projectile motion problem treated after resolving components?

As two independent one-dimensional motions.

Which motion component determines flight time on an inclined plane?

The perpendicular motion component.

Which motion component determines distance along the inclined plane?

The motion along the plane.