4.5 Relative motion in one and two dimensions  (Page 6/8)

An elephant is located on Earth’s surface at a latitude $\lambda .$ Calculate the centripetal acceleration of the elephant resulting from the rotation of Earth around its polar axis. Express your answer in terms of $\lambda ,$ the radius $R_E$ of Earth, and time T for one rotation of Earth. Compare your answer with g for $\lambda =40\text{°}.$

A proton in a synchrotron is moving in a circle of radius 1 km and increasing its speed by $v(t)=c_1+c_2{t}^2,\text{where}{c}_1=2.0\times {10}^5\text{m}\text{/}\text{s,}$
$c_2={10}^5\text{m}\text{/}{\text{s}}^3.$ (a) What is the proton’s total acceleration at t = 5.0 s? (b) At what time does the expression for the velocity become unphysical?

A propeller blade at rest starts to rotate from t = 0 s to t = 5.0 s with a tangential acceleration of the tip of the blade at $3.00\text{m}\text{/}{\text{s}}^2.$ The tip of the blade is 1.5 m from the axis of rotation. At t = 5.0 s, what is the total acceleration of the tip of the blade?

A particle is executing circular motion with a constant angular frequency of $\omega =4.00\text{rad}\text{/}\text{s}.$ If time t = 0 corresponds to the position of the particle being located at y = 0 m and x = 5 m, (a) what is the position of the particle at t = 10 s? (b) What is its velocity at this time? (c) What is its acceleration?

A particle’s centripetal acceleration is $a_{\text{C}}=4.0\text{m}\text{/}{\text{s}}^2$ at t = 0 s. It is executing uniform circular motion about an axis at a distance of 5.0 m. What is its velocity at t = 10 s?

A rod 3.0 m in length is rotating at 2.0 rev/s about an axis at one end. Compare the centripetal accelerations at radii of (a) 1.0 m, (b) 2.0 m, and (c) 3.0 m.

A particle located initially at $(1.5\widehat{j}+4.0\widehat{k})\text{m}$ undergoes a displacement of $(2.5\widehat{i}+3.2\widehat{j}-1.2\widehat{k})\text{m}\text{.}$ What is the final position of the particle?

The position of a particle is given by $\overrightarrow{r}(t)=(50\text{m}\text{/}\text{s})t\widehat{i}-(4.9\text{m}\text{/}{\text{s}}^2)t^2\widehat{j}\text{.}$ (a) What are the particle’s velocity and acceleration as functions of time? (b) What are the initial conditions to produce the motion?

A spaceship is traveling at a constant velocity of $\overrightarrow{v}(t)=250.0\widehat{i}\text{m}\text{/}\text{s}$ when its rockets fire, giving it an acceleration of $\overrightarrow{a}(t)=(3.0\widehat{i}+4.0\widehat{k})\text{m}\text{/}{\text{s}}^2.$ What is its velocity $5$ s after the rockets fire?

A crossbow is aimed horizontally at a target 40 m away. The arrow hits 30 cm below the spot at which it was aimed. What is the initial velocity of the arrow?

A long jumper can jump a distance of 8.0 m when he takes off at an angle of $45\text{°}$ with respect to the horizontal. Assuming he can jump with the same initial speed at all angles, how much distance does he lose by taking off at $30\text{°}?$

On planet Arcon, the maximum horizontal range of a projectile launched at 10 m/s is 20 m. What is the acceleration of gravity on this planet?

A mountain biker encounters a jump on a race course that sends him into the air at $60\text{°}$ to the horizontal. If he lands at a horizontal distance of 45.0 m and 20 m below his launch point, what is his initial speed?

Which has the greater centripetal acceleration, a car with a speed of 15.0 m/s along a circular track of radius 100.0 m or a car with a speed of 12.0 m/s along a circular track of radius 75.0 m?

A geosynchronous satellite orbits Earth at a distance of 42,250.0 km and has a period of 1 day. What is the centripetal acceleration of the satellite?

Two speedboats are traveling at the same speed relative to the water in opposite directions in a moving river. An observer on the riverbank sees the boats moving at 4.0 m/s and 5.0 m/s. (a) What is the speed of the boats relative to the river? (b) How fast is the river moving relative to the shore?

World’s Longest Par 3. The tee of the world’s longest par 3 sits atop South Africa’s Hanglip Mountain at 400.0 m above the green and can only be reached by helicopter. The horizontal distance to the green is 359.0 m. Neglect air resistance and answer the following questions. (a) If a golfer launches a shot that is $40\text{°}$ with respect to the horizontal, what initial velocity must she give the ball? (b) What is the time to reach the green?

When a field goal kicker kicks a football as hard as he can at $45\text{°}$ to the horizontal, the ball just clears the 3-m-high crossbar of the goalposts 45.7 m away. (a) What is the maximum speed the kicker can impart to the football? (b) In addition to clearing the crossbar, the football must be high enough in the air early during its flight to clear the reach of the onrushing defensive lineman. If the lineman is 4.6 m away and has a vertical reach of 2.5 m, can he block the 45.7-m field goal attempt? (c) What if the lineman is 1.0 m away?

A truck is traveling east at 80 km/h. At an intersection 32 km ahead, a car is traveling north at 50 km/h. (a) How long after this moment will the vehicles be closest to each other? (b) How far apart will they be at that point?