1) Suppose that a runner on a straight track covers a distance of 1 mile in exactly 4min. what was its average velocity in

(i) mih-1 (iii) cms-1
(ii) fts-1 (iv) kmh-1

2) A bullet of mass 10g traveling horizontal at a speed of 100 ms-1 embeds itself in a block of woods mass 990g suspended by a string so that it can swing freely.

Find the vertical height through which block rises.

1. These are exercises in changing dimensions. Four minutes is (1/15 mile) One mile is 4 minutes is therefore

1.0 mile/(1/15 h) = 15 mi/h

For ft/s, multiply that by
(5280 ft/mile)/(3600 sec/h), which happens to be 22/15 ft*h/s*mi, and you get 22 ft/s

Now you do the others; you need to develop the ability to do these yourself.

2. This is the classic ballistic pendulum problem. Use conservation of momentum to get the speed V of the block with embedded bullet, before swinging begins.
10*100 g m/s = V*1000 g
V = 1 m/s
Now used conservation of energy to determine how high (H) the pendulum swings.
(1/2) M V^2 = M g H
H = V^2/(2g)

1) To find the average velocity, we need to divide the distance traveled by the time taken.

(i) To convert miles to meters, we need to multiply by a conversion factor of 1609.34 meters per mile. So, 1 mile is equal to 1609.34 meters.

The average velocity in mih-1 (miles per hour) is given by:
Average Velocity = Distance / Time = 1 mile / 4 min
Converting minutes to hours, we divide by 60: 1/4 = 0.25 hours
Converting miles to meters: 1 mile * 1609.34 meters/mile = 1609.34 meters

Average Velocity in mih-1 = 1609.34 meters / 0.25 hours = 6437.36 meters per hour.

(ii) To convert meters to feet, we need to multiply by a conversion factor of 3.281 feet per meter. So, 1 meter is equal to 3.281 feet.

The average velocity in fts-1 (feet per second) is given by:
Average Velocity = Distance / Time = 1 mile / 4 min
Converting minutes to seconds, we multiply by 60: 4 min * 60 = 240 seconds
Converting miles to meters: 1 mile * 1609.34 meters/mile = 1609.34 meters
Converting meters to feet: 1609.34 meters * 3.281 feet/meter = 5280 feet

Average Velocity in fts-1 = 5280 feet / 240 seconds = 22 feet per second.

(iii) To convert miles to centimeters, we need to multiply by a conversion factor of 160934 centimeters per mile. So, 1 mile is equal to 160934 centimeters.

The average velocity in cms-1 (centimeters per second) is given by:
Average Velocity = Distance / Time = 1 mile / 4 min
Converting minutes to seconds, we multiply by 60: 4 min * 60 = 240 seconds
Converting miles to centimeters: 1 mile * 160934 centimeters/mile = 160934 centimeters

Average Velocity in cms-1 = 160934 centimeters / 240 seconds = 670.558 centimeters per second.

(iv) To convert miles to kilometers, we simply multiply by a conversion factor of 1.60934 kilometers per mile. So, 1 mile is equal to 1.60934 kilometers.

The average velocity in kmh-1 (kilometers per hour) is given by:
Average Velocity = Distance / Time = 1 mile / 4 min
Converting minutes to hours, we divide by 60: 1/4 = 0.25 hours
Converting miles to kilometers: 1 mile * 1.60934 kilometers/mile = 1.60934 kilometers

Average Velocity in kmh-1 = 1.60934 kilometers / 0.25 hours = 6.43736 kilometers per hour.

2) To find the vertical height through which the block rises, we need to consider the conservation of momentum.

Before the bullet embeds itself in the block, the bullet has momentum given by:
Momentum = Mass x Velocity
M_bullet = 10g x 100 ms-1 = 1000 gms.ms-1
Converting grams to kilograms: 10g / 1000 = 0.01 kg

After the bullet embeds itself in the block, the system now has a common velocity.
Using the conservation of momentum:
Initial momentum = Final momentum
M_bullet x V_bullet = (M_block + M_bullet) x V_final
0.01 kg x 100 ms-1 = (0.990 kg + 0.01 kg) x V_final
1 kg.m.s-1 = 1 kg x V_final

Therefore, the final velocity of the system is 1 ms-1.

To find the vertical height through which the block rises, we need to consider the conservation of mechanical energy.

The initial kinetic energy of the block is zero when it is at rest. The final kinetic energy is given by:

Final Kinetic Energy = 0.5 x M_block x V_final^2

Plugging in the values:

Final Kinetic Energy = 0.5 x 0.990 kg x (1 ms-1)^2 = 0.495 J

This final kinetic energy will be converted into potential energy as the block rises.

Potential Energy = M_block x g x h

where g is the acceleration due to gravity (approximately 9.8 ms-2), and h is the vertical height through which the block rises.

Plugging in the values:

0.495 J = 0.990 kg x 9.8 ms-2 x h

Solving for h:

h = 0.495 J / (0.990 kg x 9.8 ms-2) = 0.051 m = 5.1 cm

Therefore, the vertical height through which the block rises is 5.1 cm.

1) To find the average velocity in different units, we need to convert the given distance and time to those units.

(i) To find the average velocity in miles per hour (mih-1), we need to convert the distance from miles to kilometers and the time from minutes to hours.
1 mile = 1.60934 kilometers
4 minutes = 4/60 = 0.067 hours

Average velocity in mih-1 = distance/time = 1.60934 km / 0.067 hours

(ii) To find the average velocity in feet per second (fts-1), we need to convert the distance from miles to feet and the time from minutes to seconds.
1 mile = 5280 feet
4 minutes = 4 * 60 = 240 seconds

Average velocity in fts-1 = distance/time = 5280 feet / 240 seconds

(iii) To find the average velocity in centimeters per second (cms-1), we need to convert the distance from miles to centimeters and the time from minutes to seconds.
1 mile = 160934.4 centimeters
4 minutes = 4 * 60 = 240 seconds

Average velocity in cms-1 = distance/time = 160934.4 cm / 240 seconds

(iv) To find the average velocity in kilometers per hour (kmh-1), we don't need to convert anything because the distance is already in kilometers and the time is already in hours.
Average velocity in kmh-1 = distance/time = 1.60934 km / 0.067 hours

2) To find the vertical height through which the block rises, we need to apply the principle of conservation of momentum.

The initial momentum of the bullet before embedding itself in the block is given by:
Initial momentum = Mass of bullet * Velocity of bullet

The final momentum of the bullet-block system is zero since the block and bullet move as one after the collision.

The change in momentum is equal to the momentum of the block after the collision, which is given by the formula:
Change in momentum = Mass of block * Velocity of block

Since momentum is conserved, the initial momentum of the bullet is equal to the change in momentum of the block.

Let's denote the vertical height as h. When the block rises, the potential energy gained by the system (bullet + block) is converted from the initial kinetic energy of the bullet. Therefore, we can equate the change in momentum to the potential energy gained by the block.

Mass of bullet * Velocity of bullet = Mass of block * g * h

Here, g is the acceleration due to gravity, which is approximately 9.8 m/s^2.

Given:
Mass of bullet = 10g = 0.01kg
Velocity of bullet = 100 m/s
Mass of block = 990g = 0.99kg
Acceleration due to gravity (g) = 9.8 m/s^2

Let's substitute these values into the equation and solve for h:

0.01kg * 100 m/s = 0.99kg * 9.8 m/s^2 * h

h = (0.01kg * 100 m/s) / (0.99kg * 9.8 m/s^2)

Simplify and calculate the value of h to find the vertical height through which the block rises.