Worksheet: Relative Velocity

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In this worksheet, we will practice calculating the relative velocity of a particle with respect to another and calculating the relative velocity vector.

Q1:

A train 𝐴 of length 90 m was traveling at a speed of 170 km/h. It passed by another train 𝐡 of length 205 m. Find the time required for train 𝐴 to completely pass train 𝐡, given that train 𝐡 is moving with a speed of 152 km/h in the same direction as train 𝐴.

Q2:

A car is moving on a straight road at 84 km/h, and in the opposite direction, a motorbike is moving at 45 km/h. Suppose that the direction of the car is positive. Find the velocity of the motorbike relative to the car.

Q3:

Two cars 𝐴 and 𝐡 are moving at 142 km/h and 19 km/h in opposite directions. Given that car 𝐴 is moving in the positive direction, determine its relative velocity with respect to 𝐡.

Q4:

A ship was sailing with a uniform velocity directly toward a port that is 144 km away. A patrol aircraft passed over the ship traveling in the opposite direction at 366 km/h. When the aircraft measured the ship’s speed, it appeared to be traveling at 402 km/h. Determine the time required for the ship to reach the port.

Q5:

A person was riding a bicycle on a straight road at 14 km/h. If another person was also riding a bicycle in the same direction at 9 km/h, find the velocity of the second person with respect to the first.

Q6:

A car 𝐴 was moving in a straight line at a uniform speed π‘£οŒ . Another car 𝐡 was observed coming in the opposite direction with a relative speed of 86 km/h. When car 𝐴 doubled its speed, the relative speed became 138 km/h. If car 𝐡 was traveling with a uniform speed of π‘£οŒ‘ the whole time, find the actual speed of the two cars.

  • A 𝑣 = 3 4 /  k m h , 𝑣 = 5 2 /  k m h
  • B 𝑣 = 5 2 /  k m h , 𝑣 = 3 4 /  k m h
  • C 𝑣 = 5 2 /  k m h , 𝑣 = 1 8 /  k m h
  • D 𝑣 = 1 0 4 /  k m h , 𝑣 = 3 4 /  k m h

Q7:

A police car moving on a straight road saw a motorcycle moving in the opposite direction. It appeared to be moving at 50 km/h. At the same time, the police car saw a truck that was moving in the same direction. It appeared to be moving at 76 km/h. Determine the speed of the truck relative to the motorcycle and its direction relative to the police car.

  • A 126 km/h, in the opposite direction of the police car
  • B 26 km/h, in the same direction of the police car
  • C 26 km/h, in the opposite direction of the police car
  • D 126 km/h, in the same direction of the police car

Q8:

Two warships were moving toward each other in the same straight line at 50 km/h and 89 km/h respectively. One of the ships fired a shell toward the other when the distance to it was 4.2 km. Given that the shell’s horizontal velocity relative to the first ship was 211 km/h, how long did it take for the shell to hit the second ship?

Q9:

Two cars were moving in a straight line in opposite directions. Given that the distance between them was 17 km, the speed of one of the cars was 45 km/h, and the cars would meet after 5 minutes, determine the speed of the other car.

Q10:

A helicopter flew in a straight line at 234 km/h above a train moving in the same direction. It took the helicopter 21 seconds to travel the length of the train. Following this, the pilot halved the helicopter's speed. Given that it took the train 14 seconds to pass the helicopter traveling at this speed, find the length of the train in meters.

Q11:

A train 𝐴 travelling at 165 km/h passed another train 𝐡 that was travelling at 75 km/h in the same direction. Given that the length of train 𝐡 was 210 m and that it took 24 seconds for train 𝐴 to completely pass 𝐡, find its length 𝐿 and the time 𝑑 it would take it to cross a bridge of length 105 m.

  • A 𝐿 = 3 9 0 m , 𝑑 = 1 0 . 8 s
  • B 𝐿 = 8 9 0 m , 𝑑 = 1 0 . 8 s
  • C 𝐿 = 8 9 0 m , 𝑑 = 2 . 2 9 s
  • D 𝐿 = 3 9 0 m , 𝑑 = 2 . 2 9 s

Q12:

Two cars 𝐴 and 𝐡 are moving in the same direction on a straight road, where the velocity of car 𝐡 relative to car 𝐴 is 34 km/h. If car 𝐴 slowed down to a quarter of its velocity, the velocity of car 𝐡 relative to car 𝐴 would be 76 km/h. Determine the actual speeds of the two cars π‘£οŒ  and π‘£οŒ‘.

  • A 𝑣 = 5 6 /  k m h , 𝑣 = 9 0 /  k m h
  • B 𝑣 = 9 0 /  k m h , 𝑣 = 5 6 /  k m h
  • C 𝑣 = 5 6 /  k m h , 𝑣 = 2 0 /  k m h
  • D 𝑣 = 2 0 /  k m h , 𝑣 = 5 6 /  k m h
  • E 𝑣 = 1 4 /  k m h , 𝑣 = 2 0 /  k m h

Q13:

Two cars 𝐴 and 𝐡 are moving with velocities 41 km/h and 32 km/h, respectively, in the same direction. Determine π‘£οŒ οŒ‘.

Q14:

A police car 𝐴 moving on a straight road at 71 km/h took notice of another car 𝐡 and a bicycle 𝐢 moving on the same road. Car 𝐡 was coming from the opposite direction and appeared to be traveling at a relative speed of 181 km/h, whereas the bicycle appeared to be moving backward at a relative speed of 37 km/h. Find the actual speeds π‘£οŒ‘ of car 𝐡 and π‘£οŒ’ of bicycle 𝐢.

  • A 𝑣 = 2 5 2 /  k m h , 𝑣 = 3 4 /  k m h
  • B 𝑣 = 2 5 2 /  k m h , 𝑣 = 1 0 8 /  k m h
  • C 𝑣 = 1 1 0 /  k m h , 𝑣 = 3 4 /  k m h
  • D 𝑣 = 1 1 0 /  k m h , 𝑣 = 1 0 8 /  k m h

Q15:

A train 𝐴 of length 65 m was moving at 74 km/h when it passed another train 𝐡 of length 210 m traveling at 106 km/h in the other direction. How long did it take for train 𝐴 to completely pass train 𝐡?

Q16:

A car 𝐴 moving in a straight line measured the relative velocity of another car 𝐡 and found it to be 215 km/h in the opposite direction. After car 𝐴 reduced its speed to 0.65 of its previous value, the relative velocity of car 𝐡 became 180 km/h. Find the actual speed of both cars π‘£οŒ  and π‘£οŒ‘.

  • A 𝑣 = 5 3 . 8 5 /  k m h , 𝑣 = 2 6 8 . 8 5 /  k m h
  • B 𝑣 = 1 0 0 /  k m h , 𝑣 = 1 1 5 /  k m h
  • C 𝑣 = 5 3 . 8 5 /  k m h , 𝑣 = 1 6 1 . 1 5 /  k m h
  • D 𝑣 = 1 0 0 /  k m h , 𝑣 = 3 1 5 /  k m h

Q17:

A police car with a radar speed gun was moving on a highway at a speed of 48 km/h. A truck was moving toward it in the opposite direction. If the velocity of the truck relative to the police car was 169 km/h, determine the actual speed of the truck.

Q18:

A car, traveling at 16 km/h, and a motorcycle, traveling at 71 km/h, were moving on a straight road in the same direction. Given that n is the unit vector in the direction of the car, find the velocity of the motorcycle relative to the car.

  • A βˆ’ 5 5 n km/h
  • B βˆ’ 8 7 n km/h
  • C 5 5 n km/h
  • D 8 7 n km/h

Q19:

A police car moving in a straight line with a uniform velocity measured the relative velocity of a truck moving in the same direction to be 49 km/h. The police car then doubled its speed, measured the relative velocity again, and found that the truck appeared to be stationary. Find the real speeds of the police car 𝑣car and the truck 𝑣truck.

  • A 𝑣 = 9 8 / c a r k m h , 𝑣 = 9 8 / t r u c k k m h
  • B 𝑣 = 4 9 / c a r k m h , 𝑣 = 9 8 / t r u c k k m h
  • C 𝑣 = 9 8 / c a r k m h , 𝑣 = 4 9 / t r u c k k m h
  • D 𝑣 = 2 4 . 5 / c a r k m h , 𝑣 = 4 9 / t r u c k k m h

Q20:

City A is located at the origin, while city B is located at (300,500), where the distances are in miles. An airplane, which flies at 250 miles per hour in still air, wants to fly from city A to city B, but the wind is blowing in the direction of the positive 𝑦-axis at a speed of 50 miles per hour. Find a unit vector such that if the plane heads in this direction, it will end up at city B having flown the shortest possible distance. How long will it take to get there?

  • A 4 5 + 3 5 i j , 2.5 hours
  • B 3 5 + 4 5 i j , 2 hours
  • C 3 5 + 2 5 i j , 6 hours
  • D 2 5 + 3 5 i j , 1 hour
  • E βˆ’ 3 5 βˆ’ 4 5 i j , 4 hours

Q21:

A submarine fires a torpedo at a target as part of a training exercise. The velocity of the submarine relative to the stationary target is represented by the vector u. The velocity of the torpedo relative to the submarine is represented by the vector v. What does the vector uv+ represent?

  • Athe velocity of the torpedo relative to the stationary target
  • Bthe velocity of the submarine relative to the torpedo
  • Cthe total distance the torpedo travels
  • Dthe speed of the submarine
  • Ethe speed of the torpedo

Q22:

A fighter jet was following a bomber to shoot it down. Both planes were moving in a straight line in the same direction with the same velocity. The fighter jet fired a rocket traveling at 4,575 km/h toward the bomber. Given that the distance between the two planes was 75 km and that the rocket hit the bomber 2 minutes after it was fired, find the speed of the fighter jet.

Q23:

A certain river is half a mile wide with a current flowing at 2 miles per hour from East to West. A man swims directly toward the opposite shore from the South bank of the river at a speed of 3 miles per hour. Find the distance that he travels when swimming from shore to shore. How far down the river does he find himself when he has swam across?

  • A 1 6 √ 1 3 miles, 1 2 miles down stream
  • B 1 4 √ 1 3 miles, 1 2 miles down stream
  • C 1 6 √ 1 3 miles, 1 3 miles down stream
  • D 1 4 √ 1 3 miles, 1 3 miles down stream
  • E 1 6 √ 5 miles, 1 3 miles down stream

Q24:

A train 𝐴 of length 50 m was traveling at a speed of 116 km/h. It passed by another train 𝐡 of length 70 m. Find the time required for train 𝐴 to completely pass train 𝐡, given that train 𝐡 is moving with a speed of 98 km/h in the same direction as train 𝐴.

Q25:

A ship was sailing with a uniform velocity directly toward a port that is 98 km away. A patrol aircraft passed over the ship traveling in the opposite direction at 410 km/h. When the aircraft measured the ship’s speed, it appeared to be traveling at 459 km/h. Determine the time required for the ship to reach the port.

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