### Additional Problems

A Formula One race car is traveling at 89.0 m/s along a straight track enters a turn on the race track with radius of curvature of 200.0 m. What centripetal acceleration must the car have to stay on the track?

A particle travels in a circular orbit of radius 10 m. Its speed is changing at a rate of $15.0\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}{\text{s}}^{2}$ at an instant when its speed is 40.0 m/s. What is the magnitude of the acceleration of the particle?

The driver of a car moving at 90.0 km/h presses down on the brake as the car enters a circular curve of radius 150.0 m. If the speed of the car is decreasing at a rate of 9.0 km/h each second, what is the magnitude of the acceleration of the car at the instant its speed is 60.0 km/h?

A race car entering the curved part of the track at the Daytona 500 drops its speed from 85.0 m/s to 80.0 m/s in 2.0 s. If the radius of the curved part of the track is 316.0 m, calculate the total acceleration of the race car at the beginning and ending of reduction of speed.

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 $\lambda =40\text{\xb0}.$

A proton in a synchrotron is moving in a circle of radius 1 km and increasing its speed by $v(t)={c}_{1}+{c}_{2}{t}^{2},\phantom{\rule{0.2em}{0ex}}\text{where}\phantom{\rule{0.2em}{0ex}}{c}_{1}=2.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\text{m}\text{/}\text{s,}$

${c}_{2}={10}^{5}\text{m}\text{/}{\text{s}}^{3}.$ (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 $3.00\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}{\text{s}}^{2}.$ 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 $\omega =4.00\phantom{\rule{0.2em}{0ex}}\text{rad}\text{/}\text{s}.$ 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 ${a}_{\text{C}}=4.0\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}{\text{s}}^{2}$ at *t* = 0 s where it is on the *x*-axis and moving counterclockwise in the *xy* plane. 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 $(2.5\widehat{i}+3.2\widehat{j}-1.2\widehat{k})\phantom{\rule{0.2em}{0ex}}\text{m}\text{.}$ What is the final position of the particle?

The position of a particle is given by $\overrightarrow{r}(t)=(50\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}\text{s})t\widehat{i}-(4.9\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}{\text{s}}^{2}){t}^{2}\widehat{j}\text{.}$ (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 $45\text{\xb0}$ 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 $30\text{\xb0}?$

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 $60\text{\xb0}$ 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?