A car is heading northwest at 80 km/h and a truck is heading east at 60km/h. Determine the velocity of the truck relative to the car.
relative to car
V east = 60 + 80 cos 45
V south = 80 sin 45
To determine the velocity of the truck relative to the car, we can use vector addition.
Step 1: Convert the velocities into vector form.
The car is heading northwest at 80 km/h, which can be represented as a vector with magnitude 80 km/h and direction 45 degrees (or π/4 radians) west of north.
The truck is heading east at 60 km/h, which can be represented as a vector with magnitude 60 km/h and direction 90 degrees (or π/2 radians) west of north.
Step 2: Add the vectors together.
To add the vectors, we need to break them down into their x and y components.
For the car, the x component can be calculated as:
cos(45°) * 80 km/h = 56.57 km/h
The y component can be calculated as:
sin(45°) * 80 km/h = 56.57 km/h
For the truck, the x component is:
cos(90°) * 60 km/h = 0 km/h
The y component is:
sin(90°) * 60 km/h = 60 km/h
Adding the x components together:
56.57 km/h + 0 km/h = 56.57 km/h
Adding the y components together:
56.57 km/h + 60 km/h = 116.57 km/h
Step 3: Convert the resultant vector back into magnitude and direction.
To convert the resultant vector back into magnitude and direction, we can use the Pythagorean theorem and trigonometry.
The magnitude is calculated as:
√(56.57² + 116.57²) = 128.4 km/h (rounded to one decimal place)
The direction is calculated as:
tan^(-1)(56.57/116.57) = 25.56 degrees (rounded to two decimal places)
Therefore, the velocity of the truck relative to the car is approximately 128.4 km/h at an angle of 25.56 degrees east of north.