A police car was moving along a horizontal highway at 47 kilometers per hour. It used a radar to measure the speed of a truck moving in the same direction. Given that the reading on the radar was 50 kilometers per hour, determine the actual speed of the truck.
We’re given that the police car is moving at 47 kilometers per hour and that the velocity of the truck relative to the police car is 50 kilometers per hour. And from this information, we want to find the actual speed of the truck. We begin by letting 𝑉 𝑝 equal the velocity of the police car — that’s 47 kilometers per hour — and 𝑉 𝑡𝑝, which is the truck’s velocity read by the radar, which is 50 kilometers per hour. And remember this is the truck’s velocity relative to the police car. And since both car and truck are moving in the same direction, their velocities will have the same sign.
Now, recall that for two bodies 𝐴 and 𝐵 moving with velocities 𝑉 𝐴 and 𝑉 𝐵, the relative velocity of 𝐴 with respect to 𝐵 is 𝑉 𝐴𝐵, which is 𝑉 𝐴 minus 𝑉 𝐵. Applying this to our car and truck then, we have 𝑉 𝑡𝑝 — that’s the relative velocity of the truck with respect to the car — is equal to 𝑉 𝑡 minus 𝑉 𝑝. And that’s where 𝑉 𝑡 is the velocity of the truck.
Now we know that 𝑉 𝑡𝑝 is 50 kilometers per hour, and we know that 𝑉 𝑝 is 47 kilometers per hour. And so we have 50 is equal to 𝑉 𝑡 minus 47. And now adding 47 to both sides, on our right-hand side, negative 47 plus 47 is zero. And on our left-hand side, 50 plus 47 is 97. So, we have 𝑉 𝑡 is 97.
Now, remember, the speed is the magnitude of the velocity, so in this case that’s 97 kilometers per hour. And so the actual speed of the truck is 97 kilometers per hour.