# Problems List

##### Vectors

- If two vectors
**a**and**b**are placed so that they form the sides of a triangle( Figure 1 ), what is the vector along the third side? Let the vectors point as shown. **a**and**b**lie along the sides of a triangle, as shown in Figure 1. What is the median from their common vertex to the opposite side?- What is the vector from the vertex to the centroid of the triangle shown in the last exercise?
- Two vectors
**a**and**b**lie along two adjacent sides of a regular hexagon. In terms of these two vectors, write down the vectors that lie along the other sides. - Using vectors, show that the medians of a triangle are concurrent.
- ABCD is a parallelogram. Let us call the vector BA
**a**and the vector BC**b**. P is a point on CD such that CP : PD = 3:1. What is vector BP in terms of**a**and**b**? - For the figure in the previous exercise, write the vector from B to the intersection of the diagonal AC and the line BP(video).
- Draw various vectors in the plane and convince yourself that they can be written in terms of
**i**and**j**. - Sketch 3
**i**+ 4**j**, 3**i**– 4**j**, -2**i**+ 3**j**, -3**i**– 2**j**. - Draw a vector of length a, making an angle θ with the x-axis. Show that it is equal to a cos θ
**i**+ a sin θ**j**. - What is the length of the vector 3
**i**+ 4**j**? What angle does it make with the x-axis? - What are the lengths of the two vectors 2
**i**– 2**j**and -3**i**– 3**j**? What is the angle between them? - Show that in order to add two vectors we need to simply add their x- and y- components separately: (a
**i**+ b**j**) + (c**i**+ d**j**) = (a + c)**i**+ (b + d)**j** - Student A writes down the vector a = 4
**i**+ 3**j**in a coordinate system. Student B uses a coordinate system where the x-axis is rotated (anticlockwise) by an angle of 45 degrees with respect to the axes that student A uses. What does student B write for the same vector? - An object moves 10m at an angle of 60
^{0}south of east, then 20 m south, then 20 m at an angle of 30^{0}west of north. How far is it from its starting point? - Write down the following vectors in terms of their
**i**,**j**components: a) A vector 4 units long, making an angle of 30^{0}with the x-axis, b) A vector 10 units long, making an angle of 120^{0}with the x-axis, c) A vector 10 units long, making an angle of 240^{0}with the x-axis, d) A vector 1 unit long, making an angle of 270^{0}with the x-axis.

##### Projectile Motion

- A bullet fired with a velocity of 400 m/s at an angle of 60
^{0}with the horizon. How far up will it go and when will it land? - A ball rolls off a 1 m high table with a speed of 2 m/s. How far from the base of the table will it land?
- A cannonball is fired from the base of a 30
^{0}hill with an initial velocity of 100 m/s. The initial angle of the projectile is 60^{0}with the horizontal. How far up the hill will it land? - A plane is flying horizontally at an altitude of 200 m at a speed of 300 m/s. How far before the target should the pilot release a bomb?
- A plane flying horizontally at a height of 100 m at a speed of 120 m/s wishes to hit a tank that is moving towards it on the ground with a speed of 6 m/s. How far (horizontally) from it should it release its bomb?
- A projectile leaves the ground at an angle of 45
^{0}with a speed of 40 m/s. At a distance of 20 m from where it is launched the ground slopes away at an angle of 30^{0}to the horizontal. At what horizontal distance from its starting point will the projectile hit the ground? - The roof of a house slants down at an angle of 45
^{0}. A ball rolls down off it at a speed of 20 m/s. If the edge of the roof is 4m above the ground, how far away from it does the ball land? - A projectile is found to be having a purely horizontal velocity of 10 m/s at a height of 8m. What was its velocity when it was projected from the ground?
- A target is at a height of 2 m above the ground and 20 m away. At what point must a gun be aimed? The gun is 1m above the ground and the bullet’s speed is 300 m/s as it leaves the gun.
- Show that a bomb released from a horizontally flying plane stays directly underneath the plane as it falls if the plane maintains its velocity. This means that the pilot had better change course if she does not want the bomb going off directly under the plane. In reality, air friction slows the bomb so that it explodes behind the plane if the plane maintains its velocity. Show that the bomb stays directly beneath the plane even if the plane were climbing or diving at an angle with a constant velocity.
- A gun is aimed at a target which is at the same level. At the instant that the bullet leaves the gun the target is dropped so that it falls from rest. Show that the bullet still hits the target. Show that this happens even if the target was initially not at the same level as the gun, as long as the gun was aimed at it and the target was released at the same instant that the bullet was fired.
- A ball rolls off the top of a staircase. The stairs are 20 cm wide and 20 cm high. If it rolls off with an initial speed of 2 m/s, which step will it hit?
- A basketball player releases the ball from a height of 3m and a horizontal distance of 10m from the basket. The ball leaves her hand at a 60
^{0}angle. With what speed must the ball be released if the height of the basket is 4 m? - A steady force 20
**i**– 30**j**Newtons acts on a particle of mass 2 kg that starts out at time t = 0 with the velocity 10**i**+ 10**j**m/s. What is the particle’s displacement and velocity at the end of 4 seconds? - A 1 kg particle has the initial velocity 4
**i**+ 4**j**m/s. What steady force must act on it so that at t = 4 sec its velocity is 4**i**–4**j**m/s?

##### Circular Motion

- A 1.5 kg particle moves in a circle of radius 50 cm at the speed of 20 m/s. What is the angular velocity, the instantaneous acceleration and the net force acting on it?
- A particle moves in a circle of radius R centered at the origin. If we assume that it is located at (R,0) at t = 0, its position vector is given.Differentiate
**r**with respect to t to get the velocity as a function of time:**v**= d**r**/dt. Show that this vector is perpendicular to**r**, and hence it is tangential. Differentiate it again to get the acceleration:**a**= d**v**/dt = d^{2}**r**/dt^{2}. Show that**a**points inward along**r**and has the magnitude Rω^{2}. - Write down
**r**as a function of t for a particle moving in a circle centered at the origin, moving with a speed that is steadily increasing with time:v = a_{t}t. - A boy whirls a stone tied to a string in a horizontal circle of radius 1 m at a height of 2 m. The period of revolution is 1s. The string breaks. How far from the boy does the stone land?
- A particle moves in uniform circular motion with radius 1m, period π sec, the circle centered at the point (1,2) of the x-y coordinate system. At t = 0, it is at (2,2). The rotation is anticlockwise. What is the particle’s displacement during the first 1 sec? What is its average acceleration during this 1 sec? Find the average acceleration during the time that the particle’s radius vector rotates through an angle of 0.01 radian. Use the fact that for small angles, sin θ ~ θ and cos θ ~ 1, if needed.
- A particle moves in uniform circular motion with radius R and angular velocity ω. It is centered at (0,0). Sketch its x vs t and y vs t graphs, where x and y are coordinates.

##### Relative Motion

- Rain is falling at an angle of 30
^{0}with the vertical with a velocity of 16 m/s. With what velocity must you move in order to perceive the rain as falling vertically? To perceive it falling at an angle of 45^{0}with the vertical? - Two trains are moving on parallel tracks, with velocities of 30 m/s for train A and 20 m/s for train B in the same direction. A person in train A throws a ball with an initial velocity of 40 m/s at an angle of 60
^{0}with the horizontal, as seen in her frame. What is the velocity of the ball (magnitude and angle) as seen from train B? - A river flows at the speed of 4 m/s. A boat is capable of moving at 6 m/s in still water. In which direction must the boat be pointed in order to get to a point directly across?
- A river which flows at 6 m/s is 80 m wide. A boat is capable of going at 10 m/s in still water. What is the minimum time in which it can get to the other side? How long would it take if it had to get to a point directly across?
- A plane can fly at 300 m/s in still air. A 60 m/s wind is blowing in the north easterly direction. In which direction should the plane be pointed in order to head north? How long will it take to get to a city 400 km due north?
- An elevator is moving upward at a constant velocity of 4 m/s when an occupant drops a coin from a height of 1m. How long will the coin take to hit the ground? What if the elevator had a constant acceleration of 5 m/s
^{2}upward and the coin was dropped at the instant that the elevator was moving up with the velocity 2 m/s? - A person, standing still, finds rain coming down at a speed of 10 m/s and it is making an angle of 30° with the vertical and is eastward. A car travels along on an uphill road at a speed of 54 km per hour. If the road is along north and makes an angle of 15° with the horizontal, find the angle φ the rain makes with the vertical as seen by a person in the car, when the car is moving north( video ).