BACHELOR OF TECHNOLOGY IN RENEWABLE ENERGY & ENVIRONMENTAL PHYSICS

#TeamBtre2015

Physics problems: kinematics

Problem 1

A train covers 60 miles between 2 p.m. and 4 p.m. How fast was it going at 3 p.m.?

Solution:

The speed is traveled distance (60 miles) divided by traveled time (4pm – 2pm = 2hours):

Problem 2

Is it possible that the car could have accelerated to 55mph within 268 meters if the car can only accelerate from 0 to 60 mph in 15 seconds?

Solution:

Let us find the maximum acceleration of the car:

The car can accelerate from 0 to

in 15 seconds. Then maximum acceleration is

If the car needs to accelerate to

within 268 meters then its acceleration should be

This acceleration is less than the maximum possible acceleration, so the car can reach the speed 55 mph within 268 meters.

Problem 3

A car travels up a hill at a constant speed of 37 km/h and returns down the hill at a constant speed of 66 km/h. Calculate the average speed for the whole trip.

Solution:

By definition the average speed is the ration of the total traveled distance and the total traveled time. Let us introduce the total traveled distance of the car as L. Then the time of the travel up the hill is

The time of the travel down the hill is

The total traveled time is

Then the average velocity is

Problem 4

An archer shoots an arrow with a velocity of 30 m/s at an angle of 20 degrees with respect to the horizontal. An assistant standing on the level ground 30 m downrange from the launch point throws an apple straight up with the minimum initial speed necessary to meet the path of the arrow. What is the initial speed of the apple and at what time after the arrow is launched should the apple be thrown so that the arrow hits the apple?

Solution:

The motion of the arrow is a projectile motion. Then the motion of the arrow along horizontal direction is a motion with constant velocity:

Where the initial position is 0 and

. Then

Motion along vertical axis is a motion with constant acceleration, then

Where

, initial position is 0, and

. Then

Then we need to find the time when the arrow will be exactly above the assistant. At this moment of time x(t) =100. Then we can find the time:

Then we can find the height of the arrow at this moment of time:

The assistant should through the apple with minimal velocity so it will reach point 5.4 m and at this height the velocity should be 0. From this condition we can find the minimal velocity:

The time of the motion of the apple to this point is

Then the apple should be thrown after

Problem 5

A box sits on a horizontal wooden board. The coefficient of static friction between the box and the board is 0.5. You grab one end of the board and lift it up, keeping the other end of the board on the ground. What is the angle between the board and the horizontal direction when the box begins to slide down the board?

Solution:

The critical angle is determined by the condition:

From this equation we can find an angle

Problem 6

A 8 kg block is at rest on a horizontal floor. If you push horizontally on the 8 kg block with a force of 20 N, it just starts to move.

(a) What is the coefficient of static friction?

(b) A 10.0 kg block is stacked on top of the 8 kg block. What is the magnitude F of the force, acting horizontally on the 8 kg block as before, that is required to make the two blocks start to move?

Solution:

The magnitude of horizontal force should be equal to the magnitude of the maximal static friction force, which is equal to the product of the coefficient of static friction and the normal force (gravitation force in the present problem).

(a) The gravitation force is mg=8*9.8 = 78.4 N. Then the coefficient of static friction is

(b) Now we know the coefficient of static friction and we know the normal force: 18*9.8 = 176.4 N. Then we can find the magnitude of force F:

Problem 7

A car is accelerating at

. Find its acceleration in

Solution:

To find an acceleration in

we need to use the relations:

Then we can write:

Problem 8.

We drive a distance of 1 km at 16 km/h. Then we drive an additional distance of 1 km at 32 km/h. What is our average speed?

Solution:

By definition the average speed is the ratio of traveled distance and traveled time.

The traveled distance is 2 km.

The traveled time is the sum of two contributions:

• time of the motion a distance 1 km with speed 16 km/h. It is

• time of the motion a distance 1 km with speed 32 km/h. It is

Then the average speed is

Problem 9.

An airliner reaches its takeoff speed of 163 mph in 36.2 s. What is the magnitude of its average acceleration.

Solution:

By definition the acceleration is the ratio of the change of velocity and the traveled time. In the present problem the change of velocity is 163 mph = 0.447*163 m/s = 72.9 m/s. The traveled time is 36.2 s.

Then the average acceleration is