PY205 Ch3 Questions

Two cannonballs, A and B, are fired from the ground with identical initial speeds, but with θA larger than θB. (a) Which cannonball reaches a higher elevation? (b) Which stays longer in the air? (c) Which travels farther?

(a) Cannonball A, because it has a larger initial vertical velocity component.(b) Cannonball A, same reason. (c) It depends. If θA < 45o, cannonball A will travel farther. If θB > 45o, cannonball B will travel farther. If θA > 45o and θB < 45o, the cannonball whose angle is closest to 45o will travel farther.

A person sitting in an enclosed train car, moving at constant velocity, throws a ball straight up into the air in her refer ence frame, (a) Where does the ball land? What is your answer if the car (b) accelerates, (c) decelerates, (d) rounds a curve, (e) moves with constant velocity but is open to the air?

(a) The ball lands back in her hand (b) The ball lands behind her hand. (c) The ball lands in front of her hand. (d) The ball lands beside her hand, to the outside of the curve. (e) The ball lands behind her hand, if air resistance is not negligible.

Can the magnitude of a vector ever (a) equal, or (b) be less than, one of its components?

(a) Yes. In three dimensions, the magnitude of a vector is the square root of the sum of the squares of the components. If two of the components are zero, the magnitude of the vector is equal to the magnitude of the remaining component. (b) No.

A projectile has the least speed at what point in its path?

A projectile has the least speed at the top of its path. At that point the vertical speed is zero. The horizontal speed remains constant throughout the flight, if we neglect the effects of air resistance.

A projectile is launched at an upward angle of 30° to the horizontal with a speed of 30 m/s. How does the horizontal component of its velocity 1.0 s after launch compare with its horizontal component of velocity 2.0 s after launch, ignoring air resistance?

As long as air resistance is negligible, the horizontal component of the projectile's velocity remains constant until it hits the ground. It is in the air longer than 2.0 s, so the value of the horizontal component of its velocity at 1.0 s and 2.0 s is the same.

If you stand motionless under an umbrella in a rainstorm where the drops fall vertically you remain relatively dry. However, if you start running, the rain begins to hit your legs even if they remain under the umbrella. Why?

As you run forward, the umbrella also moves forward and stops raindrops that are at its height above the ground. Raindrops that have already passed the height of the umbrella continue to move toward the ground unimpeded. As you run, you move into the space where the raindrops are continuing to fall (below the umbrella). Some of them will hit your legs and you will get wet.

Can you give several examples of an object's motion in which a great distance is traveled but the displacement is zero?

Automobile races that begin and end at the same place; a round-trip by car from New York to San Francisco and back; a balloon flight around the world.

It was reported in World War I that a pilot flying at an altitude of 2km caught in his bare hands a bullet fired at the plane! Using the fact that a bullet slows down considerably due to air resistance, explain how this incident occurred.

If the bullet was fired from the ground, then the y-component of its velocity slowed considerably by the time it reached an altitude of 2.0 km, because of both acceleration due to gravity (downward) and air resistance. The x-component of its velocity would have slowed due to air resistance as well. Therefore, the bullet could have been traveling slowly enough to be caught!

A child wishes to determine the speed a slingshot imparts to a rock. How can this be done using only a meter stick, a rock, and the slingshot?

Launch the rock with a horizontal velocity from a known height over level ground. Use the equations for projectile motion in the y-direction to find the time the rock is in the air. (Note that the initial velocity has a zero y-component.) Use this time and the horizontal distance the rock travels in the equation for x-direction projectile motion to find the speed in the x-direction, which is the speed the slingshot imparts. The meter stick is used to measure the initial height and the horizontal distance the rock travels.

In archery, should the arrow be aimed directly at the target? How should your angle of aim depend on the distance to the target?

No. The arrow will fall toward the ground as it travels toward the target, so it should be aimed above the target. Generally, the farther you are from the target, the higher above the target the arrow should be aimed, up to a maximum launch angle of 45o. (The maximum range of a projectile that starts and stops at the same height occurs when the launch angle is 45o.)

Can you conclude that a car is not accelerating if its speedometer indicates a steady 60 km/h?

No. The car may be traveling at a constant speed of 60 km/h and going around a curve, in which case it would be accelerating.

Can two vectors, of unequal magnitude, add up to give the zero vector? Can three unequal vectors? Under what conditions?

No. The only way that two vectors can add up to give the zero vector is if they have the same magnitude and point in exactly opposite directions. However, three vectors of unequal magnitudes can add up to the zero vector. As a one-dimensional example, a vector 10 units long in the positive x direction added to two vectors of 4 and 6 units each in the negative x direction will result in the zero vector. In two dimensions, consider any three vectors that when added form a triangle.

One car travels due east at 40 km/h, and a second car travels north at 40 km/h. Are their velocities equal? Explain.

No. Velocity is a vector quantity, with a magnitude and direction. If two vectors have different directions, they cannot be equal.

Can the displacement vector for a particle moving in two dimensions ever be longer than the length of path traveled by the particle over the same time interval? Can it ever be less? Discuss.

The length of the displacement vector is the straight-line distance between the beginning point and the ending point of the trip and therefore the shortest distance between the two points. If the path is a straight line, then the length of the displacement vector is the same as the length of the path. If the path is curved or consists of different straight line segments, then the distance from beginning to end will be less than the path length. Therefore, the displacement vector can never be longer than the length of the path traveled, but it can be shorter.

Two vectors have length V1 = 3.5 km and V2 = 4.0 km. What are the maximum and minimum magnitudes of their vector sum?

The maximum magnitude of the sum is 7.5 km, in the case where the vectors are parallel. The minimum magnitude of the sum is 0.5 km, in the case where the vectors are antiparallel.

Does the odometer of a car measure a scalar or a vector quantity? What about the speedometer?

The odometer and the speedometer of the car both measure scalar quantities (distance and speed, respectively).

During baseball practice, a batter hits a very high fly ball and then runs in a straight line and catches it. Which had the greater displacement, the player or the ball?

The player and the ball have the same displacement.

Two rowers, who can row at the same speed in still water, set off across a river at the same time. One heads straight across and is pulled downstream somewhat by the current. The other one heads upstream at an angle so as to arrive at a point opposite the starting point. Which rower reaches the opposite side first?

The time it takes to cross the river depends on the component of velocity in the direction straight across the river. Imagine a river running to the east and rowers beginning on the south bank. Let the still water speed of both rowers be v. Then the rower who heads due north (straight across the river) has a northward velocity component v. The rower who heads upstream, though, has a northward velocity component of less than v. Therefore, the rower heading straight across reaches the opposite shore first. (However, she won't end up straight across from where she started!)

If you are riding on a train that speeds past another train moving in the same direction on an adjacent track, it appears that the other train is moving backward. Why?

This is a question of relative velocity. From the point of view of an observer on the ground, both trains are moving in the same direction (forward), but at different speeds. From your point of view on the faster train, the slower train (and the ground) will appear to be moving backward. (The ground will be moving backward faster than the slower train!)

If V = V1 + V2, is V necessarily greater than V1 and/or V2? Discuss.

V is the magnitude of the vector V ; it is not necessarily larger than the magnitudes V1 and V2. For instance, if V1 and V2 have the same magnitude as each other and are in opposite directions, then V 12 is zero.

Can a particle with constant speed be accelerating? What if it has constant velocity?

Yes. A particle traveling around a curve while maintaining a constant speed is accelerating because its direction is changing. A particle with a constant velocity cannot be accelerating, since the velocity is not changing in magnitude or direction.