A motorcycle traveling 91.0 km/hr approaches a car traveling in the same direction at 83.0 km/hr. When the motorcycle is 52.0 m behind the car, the rider accelerates and passes the car 16.0 s later. What is the acceleration of the motorcycle (in meters/second^2)?

.28

To find the acceleration of the motorcycle, we first need to determine the initial velocity (u) and final velocity (v) of the motorcycle.

1. Convert the given speeds from km/hr to m/s:
- Speed of the motorcycle (u) = 91.0 km/hr = 91.0 × (1000/3600) m/s = 25.28 m/s
- Speed of the car (v) = 83.0 km/hr = 83.0 × (1000/3600) m/s = 23.06 m/s

2. Calculate the relative velocity (v_rel) between the motorcycle and the car:
- v_rel = v - u
- v_rel = 23.06 m/s - 25.28 m/s
- v_rel = -2.22 m/s

Note: The negative sign indicates that the car is moving faster than the motorcycle.

3. Calculate the relative distance (s_rel) between the motorcycle and the car:
- s_rel = 52.0 m

4. Use the formula for relative motion:
- s_rel = v_rel * t + (1/2) * a * t^2
- where:
- s_rel is the relative distance (52.0 m),
- v_rel is the relative velocity (-2.22 m/s),
- t is the time interval (16.0 s),
- a is the acceleration (to be determined).

5. Rearrange the formula to solve for acceleration:
- s_rel = v_rel * t + (1/2) * a * t^2
- 52.0 m = (-2.22 m/s)(16.0 s) + (1/2) * a * (16.0 s)^2

6. Solve for acceleration (a):
- 52.0 m = (-2.22 m/s)(16.0 s) + (1/2) * a * (16.0 s)^2
- 52.0 m = -35.52 m + 128.0 a
- 80.52 m = 128.0 a
- a = 80.52 m / 128.0
- a ≈ 0.6289 m/s^2

Therefore, the acceleration of the motorcycle is approximately 0.6289 m/s^2.

To find the acceleration of the motorcycle, we need to use the equations of motion. We can start by finding the initial velocity of the motorcycle.

Given:
Speed of the motorcycle = 91.0 km/hr
Speed of the car = 83.0 km/hr
Distance between motorcycle and car = 52.0 m
Time taken for the motorcycle to pass the car = 16.0 s

Step 1: Convert the speeds to meters per second (m/s).
Speed of the motorcycle = 91.0 km/hr x (1000 m/1 km) x (1 hr/3600 s) ≈ 25.28 m/s
Speed of the car = 83.0 km/hr x (1000 m/1 km) x (1 hr/3600 s) ≈ 23.06 m/s

Step 2: Calculate the initial velocity of the motorcycle.
Initial velocity of the motorcycle = final velocity of the motorcycle - relative velocity of the car
Relative velocity of the car = speed of the car - speed of the motorcycle
Relative velocity of the car = 23.06 m/s - 25.28 m/s = -2.22 m/s (negative sign indicates that the car is moving ahead)

Initial velocity of the motorcycle = 25.28 m/s - (-2.22 m/s) = 27.50 m/s

Step 3: Calculate the acceleration of the motorcycle.
Acceleration = (final velocity - initial velocity) / time taken
Final velocity of the motorcycle = distance traveled / time taken
Distance traveled = Distance between motorcycle and car + length of the car
Distance traveled = 52.0 m + length of the car (considering the car's length as negligible in this case)

Final velocity of the motorcycle = (Total distance traveled) / time taken
Total distance traveled = Distance between motorcycle and car + Length of the car
Total distance traveled = 52.0 m + length of the car

Substituting the values:
Final velocity of the motorcycle = (52.0 m + length of the car) / 16.0 s

Acceleration = [(52.0 m + length of the car) / 16.0 s - 27.50 m/s] / 16.0 s

Note: To get the length of the car, we would need additional information.

So, the equation for acceleration of the motorcycle contains a variable (length of the car) for which we need more information. Therefore, we cannot determine the exact acceleration of the motorcycle without knowing the length of the car.

Easy way: relative motion

relative initial speed: (91-83)km/hr=8km/hr= 2.2222 m/s

distancerelative=virelative*time+1/2 a t^2

52=2.22*16+1/2 a (256)

solve for a.

Harder method:
find the distance the slow car goes in 16 seconds: (velocity*time)
find the distance the fast car must go (above+ 52)
then
that distance=Vi*16 + 1/2 *256 a
check my thinkikng.