Derivatives and Rate of Change Word Problems: Motion and Relative Speed

1.A body has an a equation of motion measured in metres after t seconds such that
s=4t3-14t2+40t+8.

a) When and where is the body momentarily at rest?
b) For what time interval is it moving forward?
c) During what times is its acceleration negative?
d) Draw three separate graphs for acceleration, velocity, and displacement. Explain the relationship between the important information on the graphs.

2.Mrs. Gordon lives on an island 1 km from the mainland. She paddles her canoe at 3km/h and jogs at 5km/h. The nearest drug store is 3 km along the shore from the point on the shore closest to the island. Where should she land to reach the drug store in minimum time?

3.A power boat travels west at 24 km/h. At the instant it passes a buoy, a sailboat sailing north at 7 km/h is 25 km south of the buoy. Calculate the positions of the vessels when there is a minimum distance between them.

Motion and Relative Speed are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

A child in an airport is able to cover 270 meters in 3 minutes running at a speed down a moving sidewalk in the direction of the sidewalks motion. Running at the same speed in the direction opposite to the sidewalk's movement, the child is able to cover 256 meters in 4 mintues. What is the child's running speedand the she spe

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1. The launch catapult of an aircraft carrier gives the 7Mg jet airplane gives the airplane a constant acceleration and launches the airplane in a distance of 100 m measured along the angled takeoff ramp. The carrier is moving at a constant speed of 16 m/s. If an ab

A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P?

A boat travels up a river a distance d and then comes back to the origin.
The maximum speed of the boat relative to the water is v. And the river has scalar speed u (u < v).
What is the minimum time for this trip?

Early one morning it began to snow at a constant rate. At 7 AM a snowplow set off to clear a road. By 8 AM it had traveled 2 miles but it took two more hours for the snowplow to go another 2 miles. Assuming that the snowplow clears snow from the road at a constant rate (in cubic feet per hour), at what time did it start to snow?

An athlete executing a long jump leaves the ground at a 30.8° angle and travels 7.61 m with a take-off speed of 9.21 m/s. If this speed were increased by just 4.1%, how much longer would the jump be?

Kim starts to walk 3 mi to school at 7:30 a.m. with a temperature of 0 degrees F. Her brother Bryan starts at 7:45 a.m. on his bicycle, traveling 10 mph faster than Kim. They they get to school at the same time, then how fast is each one traveling?

A child in an airport is able to cover 392 meters in 4 minutes running at a steady speed down a moving sidewalk in the direction of the sidewalk's motion. Running at the same speed in the direction opposite to the sidewalk's movement, the child is able to cover 350 meters in 5 minutes. What is the child's running speed on a stil

At time t, the mass of a rocket is M(1-kt), where M and k are constants. At time t, the rocket is moving with speed v vertically upwards near the Earth's surface against constant gravity. Burnt fuel is expelled vertically downwards at speed u relative to the rocket.
a) Show that (1-kt)dv/dt = ku-g(1-kt)
b) Given that v=0 when